Misplaced Pages

#890109

93-562: The coefficient a in the term of ax y is known as the binomial coefficient ( n b ) {\displaystyle {\tbinom {n}{b}}} or ( n c ) {\displaystyle {\tbinom {n}{c}}} (the two have the same value). These coefficients for varying n and b can be arranged to form Pascal's triangle . These numbers also occur in combinatorics , where ( n b ) {\displaystyle {\tbinom {n}{b}}} gives

186-441: A k ( t k ) {\textstyle \sum _{k=0}^{d}a_{k}{\binom {t}{k}}} of binomial coefficients, because the binomial coefficients consist of one polynomial of each degree. The coefficient a k is the k th difference of the sequence p (0), p (1), ..., p ( k ). Explicitly, Each polynomial ( t k ) {\displaystyle {\tbinom {t}{k}}}

279-489: A i is a nonnegative integer is given by ⁠ ( n + k − 1 n − 1 ) {\displaystyle {\tbinom {n+k-1}{n-1}}} ⁠ . Most of these interpretations can be shown to be equivalent to counting k -combinations. Several methods exist to compute the value of ( n k ) {\displaystyle {\tbinom {n}{k}}} without actually expanding

372-532: A + b ) ( n ) = ∑ k = 0 n ( n k ) a ( n − k ) b ( k ) . {\displaystyle (a+b)^{(n)}=\sum _{k=0}^{n}{\binom {n}{k}}a^{(n-k)}b^{(k)}.} The case c = 0 recovers the usual binomial theorem. More generally, a sequence { p n } n = 0 ∞ {\displaystyle \{p_{n}\}_{n=0}^{\infty }} of polynomials

465-472: A and b , the binomial theorem with n = 2 is the geometrically evident fact that a square of side a + b can be cut into a square of side a , a square of side b , and two rectangles with sides a and b . With n = 3 , the theorem states that a cube of side a + b can be cut into a cube of side a , a cube of side b , three a × a × b rectangular boxes, and three a × b × b rectangular boxes. In calculus , this picture also gives

558-827: A binomial power or counting k -combinations. One method uses the recursive , purely additive formula ( n k ) = ( n − 1 k − 1 ) + ( n − 1 k ) {\displaystyle {\binom {n}{k}}={\binom {n-1}{k-1}}+{\binom {n-1}{k}}} for all integers n , k {\displaystyle n,k} such that 1 ≤ k < n , {\displaystyle 1\leq k<n,} with boundary values ( n 0 ) = ( n n ) = 1 {\displaystyle {\binom {n}{0}}={\binom {n}{n}}=1} for all integers n ≥ 0 . The formula follows from considering

651-404: A characteristic 0 field K , a polynomial in K [ t ] takes values in R at all integers if and only if it is an R -linear combination of binomial coefficient polynomials. The integer-valued polynomial 3 t (3 t + 1) / 2 can be rewritten as The factorial formula facilitates relating nearby binomial coefficients. For instance, if k is a positive integer and n is arbitrary, then and, with

744-616: A fixed set of n elements. For example, there are ( 4 2 ) = 6 {\displaystyle {\tbinom {4}{2}}=6} ways to choose 2 elements from {1, 2, 3, 4} , namely {1, 2} , {1, 3} , {1, 4} , {2, 3} , {2, 4} and {3, 4} . The first form of the binomial coefficients can be generalized to ( z k ) {\displaystyle {\tbinom {z}{k}}} for any complex number z and integer k ≥ 0 , and many of their properties continue to hold in this more general form. Andreas von Ettingshausen introduced

837-536: A generalization of the binomial formula (with one of the variables set to 1), which justifies still calling the ( α k ) {\displaystyle {\tbinom {\alpha }{k}}} binomial coefficients: This formula is valid for all complex numbers α and X with | X | < 1. It can also be interpreted as an identity of formal power series in X , where it actually can serve as definition of arbitrary powers of power series with constant coefficient equal to 1;

930-426: A geometric proof of the derivative ( x n ) ′ = n x n − 1 : {\displaystyle (x^{n})'=nx^{n-1}:} if one sets a = x {\displaystyle a=x} and b = Δ x , {\displaystyle b=\Delta x,} interpreting b as an infinitesimal change in a , then this picture shows

1023-424: A little more work, We can also get Moreover, the following may be useful: For constant n , we have the following recurrence: To sum up, we have The formula says that the elements in the n th row of Pascal's triangle always add up to 2 raised to the n th power. This is obtained from the binomial theorem ( ∗ ) by setting x = 1 and y = 1 . The formula also has a natural combinatorial interpretation:

SECTION 10

#1732780911891

1116-462: A problem when evaluated at integers from 0 {\displaystyle 0} to t − 1 {\displaystyle t-1} , but using identities below we can compute the derivative as: Over any field of characteristic 0 (that is, any field that contains the rational numbers ), each polynomial p ( t ) of degree at most d is uniquely expressible as a linear combination ∑ k = 0 d

1209-528: A specific negative value of y : ( x − 2 ) 3 = x 3 − 3 x 2 ( 2 ) + 3 x ( 2 ) 2 − 2 3 = x 3 − 6 x 2 + 12 x − 8. {\displaystyle {\begin{aligned}(x-2)^{3}&=x^{3}-3x^{2}(2)+3x(2)^{2}-2^{3}\\&=x^{3}-6x^{2}+12x-8.\end{aligned}}} For positive values of

1302-487: A specific positive value of y : ( x + 2 ) 3 = x 3 + 3 x 2 ( 2 ) + 3 x ( 2 ) 2 + 2 3 = x 3 + 6 x 2 + 12 x + 8. {\displaystyle {\begin{aligned}(x+2)^{3}&=x^{3}+3x^{2}(2)+3x(2)^{2}+2^{3}\\&=x^{3}+6x^{2}+12x+8.\end{aligned}}} A simple example with

1395-828: A symmetry that is less evident from the multiplicative formula (though it is from the definitions) which leads to a more efficient multiplicative computational routine. Using the falling factorial notation , ( n k ) = { n k _ / k ! if    k ≤ n 2 n n − k _ / ( n − k ) ! if    k > n 2 . {\displaystyle {\binom {n}{k}}={\begin{cases}n^{\underline {k}}/k!&{\text{if }}\ k\leq {\frac {n}{2}}\\n^{\underline {n-k}}/(n-k)!&{\text{if }}\ k>{\frac {n}{2}}\end{cases}}.} The multiplicative formula allows

1488-495: A work by Al-Karaji , quoted by Al-Samaw'al in his "al-Bahir". Al-Karaji described the triangular pattern of the binomial coefficients and also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using an early form of mathematical induction . The Persian poet and mathematician Omar Khayyam was probably familiar with the formula to higher orders, although many of his mathematical works are lost. The binomial expansions of small degrees were known in

1581-464: Is integer-valued : it has an integer value at all integer inputs t {\displaystyle t} . (One way to prove this is by induction on k using Pascal's identity .) Therefore, any integer linear combination of binomial coefficient polynomials is integer-valued too. Conversely, ( 4 ) shows that any integer-valued polynomial is an integer linear combination of these binomial coefficient polynomials. More generally, for any subring R of

1674-431: Is a natural number for all integer n ≥ 0 and all integer k , a fact that is not immediately obvious from formula (1) . To the left and right of Pascal's triangle, the entries (shown as blanks) are all zero. Pascal's rule also gives rise to Pascal's triangle : Row number n contains the numbers ( n k ) {\displaystyle {\tbinom {n}{k}}} for k = 0, …, n . It

1767-488: Is a nonnegative integer n , then all terms with k > n are zero, and the infinite series becomes a finite sum, thereby recovering the binomial formula. However, for other values of α , including negative integers and rational numbers, the series is really infinite. Pascal's rule is the important recurrence relation which can be used to prove by mathematical induction that ( n k ) {\displaystyle {\tbinom {n}{k}}}

1860-807: Is a nonnegative integer, the binomial coefficients for k > r are zero, so this equation reduces to the usual binomial theorem, and there are at most r + 1 nonzero terms. For other values of r , the series typically has infinitely many nonzero terms. For example, r = 1/2 gives the following series for the square root: 1 + x = 1 + 1 2 x − 1 8 x 2 + 1 16 x 3 − 5 128 x 4 + 7 256 x 5 − ⋯ . {\displaystyle {\sqrt {1+x}}=1+{\frac {1}{2}}x-{\frac {1}{8}}x^{2}+{\frac {1}{16}}x^{3}-{\frac {5}{128}}x^{4}+{\frac {7}{256}}x^{5}-\cdots .} Taking r = −1 ,

1953-610: Is a positive integer known as a binomial coefficient , defined as ( n k ) = n ! k ! ( n − k ) ! = n ( n − 1 ) ( n − 2 ) ⋯ ( n − k + 1 ) k ( k − 1 ) ( k − 2 ) ⋯ 2 ⋅ 1 . {\displaystyle {\binom {n}{k}}={\frac {n!}{k!\,(n-k)!}}={\frac {n(n-1)(n-2)\cdots (n-k+1)}{k(k-1)(k-2)\cdots 2\cdot 1}}.} This formula

SECTION 20

#1732780911891

2046-508: Is also a polynomial in x and y , and [ ( x + y ) n + 1 ] j , k = [ ( x + y ) n ] j − 1 , k + [ ( x + y ) n ] j , k − 1 , {\displaystyle [(x+y)^{n+1}]_{j,k}=[(x+y)^{n}]_{j-1,k}+[(x+y)^{n}]_{j,k-1},} since if j + k = n + 1 , then ( j − 1) + k = n and j + ( k − 1) = n . Now,

2139-635: Is also referred to as the binomial formula or the binomial identity . Using summation notation , it can be written more concisely as ( x + y ) n = ∑ k = 0 n ( n k ) x n − k y k = ∑ k = 0 n ( n k ) x k y n − k . {\displaystyle (x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}=\sum _{k=0}^{n}{n \choose k}x^{k}y^{n-k}.} The final expression follows from

2232-542: Is binomial if and only if its basis operator is a Delta operator . Writing E a {\displaystyle E^{a}} for the shift by a {\displaystyle a} operator, the Delta operators corresponding to the above "Pochhammer" families of polynomials are the backward difference I − E − c {\displaystyle I-E^{-c}} for c > 0 {\displaystyle c>0} ,

2325-524: Is constructed by first placing 1s in the outermost positions, and then filling each inner position with the sum of the two numbers directly above. This method allows the quick calculation of binomial coefficients without the need for fractions or multiplications. For instance, by looking at row number 5 of the triangle, one can quickly read off that Binomial coefficients are of importance in combinatorics because they provide ready formulas for certain frequent counting problems: For any nonnegative integer k ,

2418-1044: Is given by the formula ( n k ) = n ! k ! ( n − k ) ! , {\displaystyle {\binom {n}{k}}={\frac {n!}{k!\;(n-k)!}},} which is defined in terms of the factorial function n ! . Equivalently, this formula can be written ( n k ) = n ( n − 1 ) ⋯ ( n − k + 1 ) k ( k − 1 ) ⋯ 1 = ∏ ℓ = 1 k n − ℓ + 1 ℓ = ∏ ℓ = 0 k − 1 n − ℓ k − ℓ {\displaystyle {\binom {n}{k}}={\frac {n(n-1)\cdots (n-k+1)}{k(k-1)\cdots 1}}=\prod _{\ell =1}^{k}{\frac {n-\ell +1}{\ell }}=\prod _{\ell =0}^{k-1}{\frac {n-\ell }{k-\ell }}} with k factors in both

2511-432: Is in combinatorics, where it gives the number of ways, disregarding order, that k objects can be chosen from among n objects; more formally, the number of k -element subsets (or k - combinations ) of an n -element set. This number can be seen as equal to the one of the first definition, independently of any of the formulas below to compute it: if in each of the n factors of the power (1 + X ) one temporarily labels

2604-591: Is invertible, and ‖ y / x ‖ < 1 . A version of the binomial theorem is valid for the following Pochhammer symbol -like family of polynomials: for a given real constant c , define x ( 0 ) = 1 {\displaystyle x^{(0)}=1} and x ( n ) = ∏ k = 1 n [ x + ( k − 1 ) c ] {\displaystyle x^{(n)}=\prod _{k=1}^{n}[x+(k-1)c]} for n > 0. {\displaystyle n>0.} Then (

2697-978: Is obtained by substituting 1 for y , so that it involves only a single variable . In this form, the formula reads ( x + 1 ) n = ( n 0 ) x 0 + ( n 1 ) x 1 + ( n 2 ) x 2 + ⋯ + ( n n − 1 ) x n − 1 + ( n n ) x n = ∑ k = 0 n ( n k ) x k . ) {\displaystyle {\begin{aligned}(x+1)^{n}&={n \choose 0}x^{0}+{n \choose 1}x^{1}+{n \choose 2}x^{2}+\cdots +{n \choose {n-1}}x^{n-1}+{n \choose n}x^{n}\\[4mu]&=\sum _{k=0}^{n}{n \choose k}x^{k}.{\vphantom {\Bigg )}}\end{aligned}}} Here are

2790-997: Is often defined by the formula e = lim n → ∞ ( 1 + 1 n ) n . {\displaystyle e=\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}.} Applying the binomial theorem to this expression yields the usual infinite series for e . In particular: ( 1 + 1 n ) n = 1 + ( n 1 ) 1 n + ( n 2 ) 1 n 2 + ( n 3 ) 1 n 3 + ⋯ + ( n n ) 1 n n . {\displaystyle \left(1+{\frac {1}{n}}\right)^{n}=1+{n \choose 1}{\frac {1}{n}}+{n \choose 2}{\frac {1}{n^{2}}}+{n \choose 3}{\frac {1}{n^{3}}}+\cdots +{n \choose n}{\frac {1}{n^{n}}}.} Binomial coefficient In mathematics ,

2883-425: Is readily derived by evaluating C n , k / C n , k − 1 {\displaystyle C_{n,k}/C_{n,k-1}} and can intuitively be understood as starting at the leftmost coefficient of the n {\displaystyle n} -th row of Pascal's triangle , whose value is always 1 {\displaystyle 1} , and recursively computing

Binomial theorem - Misplaced Pages Continue

2976-408: Is related to binomials for the following reason: if we write ( x + y ) as a product ( x + y ) ( x + y ) ( x + y ) ⋯ ( x + y ) , {\displaystyle (x+y)(x+y)(x+y)\cdots (x+y),} then, according to the distributive law , there will be one term in the expansion for each choice of either x or y from each of

3069-739: Is said to be of binomial type if An operator Q {\displaystyle Q} on the space of polynomials is said to be the basis operator of the sequence { p n } n = 0 ∞ {\displaystyle \{p_{n}\}_{n=0}^{\infty }} if Q p 0 = 0 {\displaystyle Qp_{0}=0} and Q p n = n p n − 1 {\displaystyle Qp_{n}=np_{n-1}} for all n ⩾ 1 {\displaystyle n\geqslant 1} . A sequence { p n } n = 0 ∞ {\displaystyle \{p_{n}\}_{n=0}^{\infty }}

3162-1175: Is the Pochhammer symbol , here standing for a falling factorial . This agrees with the usual definitions when r is a nonnegative integer. Then, if x and y are real numbers with | x | > | y | , and r is any complex number, one has ( x + y ) r = ∑ k = 0 ∞ ( r k ) x r − k y k = x r + r x r − 1 y + r ( r − 1 ) 2 ! x r − 2 y 2 + r ( r − 1 ) ( r − 2 ) 3 ! x r − 3 y 3 + ⋯ . {\displaystyle {\begin{aligned}(x+y)^{r}&=\sum _{k=0}^{\infty }{r \choose k}x^{r-k}y^{k}\\&=x^{r}+rx^{r-1}y+{\frac {r(r-1)}{2!}}x^{r-2}y^{2}+{\frac {r(r-1)(r-2)}{3!}}x^{r-3}y^{3}+\cdots .\end{aligned}}} When r

3255-428: Is the coefficient of the x term. Arranging the numbers ( n 0 ) , ( n 1 ) , … , ( n n ) {\displaystyle {\tbinom {n}{0}},{\tbinom {n}{1}},\ldots ,{\tbinom {n}{n}}} in successive rows for n = 0, 1, 2, ... gives a triangular array called Pascal's triangle , satisfying

3348-452: Is the compact form, often used in proofs and derivations, which makes repeated use of the familiar factorial function: ( n k ) = n ! k ! ( n − k ) ! for    0 ≤ k ≤ n , {\displaystyle {\binom {n}{k}}={\frac {n!}{k!\,(n-k)!}}\quad {\text{for }}\ 0\leq k\leq n,} where n ! denotes

3441-488: Is the inductive hypothesis with n + 1 substituted for n and so completes the inductive step. Around 1665, Isaac Newton generalized the binomial theorem to allow real exponents other than nonnegative integers. (The same generalization also applies to complex exponents.) In this generalization, the finite sum is replaced by an infinite series . In order to do this, one needs to give meaning to binomial coefficients with an arbitrary upper index, which cannot be done using

3534-427: The C stands for combinations or choices ; the C notation means the number of ways to choose k out of n objects. Many calculators use variants of the C notation because they can represent it on a single-line display. In this form the binomial coefficients are easily compared to the numbers of k -permutations of n , written as P ( n , k ) , etc. For natural numbers (taken to include 0) n and k ,

3627-461: The binomial coefficients are the positive integers that occur as coefficients in the binomial theorem . Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written ( n k ) . {\displaystyle {\tbinom {n}{k}}.} It is the coefficient of the x term in the polynomial expansion of the binomial power (1 + x ) ; this coefficient can be computed by

3720-512: The complex numbers the binomial theorem can be combined with de Moivre's formula to yield multiple-angle formulas for the sine and cosine . According to De Moivre's formula, cos ⁡ ( n x ) + i sin ⁡ ( n x ) = ( cos ⁡ x + i sin ⁡ x ) n . {\displaystyle \cos \left(nx\right)+i\sin \left(nx\right)=\left(\cos x+i\sin x\right)^{n}.} Using

3813-496: The definition of the derivative via a difference quotient and taking limits means that the higher order terms, ( Δ x ) 2 {\displaystyle (\Delta x)^{2}} and higher, become negligible, and yields the formula ( x n ) ′ = n x n − 1 , {\displaystyle (x^{n})'=nx^{n-1},} interpreted as If one integrates this picture, which corresponds to applying

Binomial theorem - Misplaced Pages Continue

3906-648: The fundamental theorem of calculus , one obtains Cavalieri's quadrature formula , the integral ∫ x n − 1 d x = 1 n x n {\displaystyle \textstyle {\int x^{n-1}\,dx={\tfrac {1}{n}}x^{n}}} – see proof of Cavalieri's quadrature formula for details. The coefficients that appear in the binomial expansion are called binomial coefficients . These are usually written ( n k ) , {\displaystyle {\tbinom {n}{k}},} and pronounced " n choose k ". The coefficient of x y

3999-479: The n faces, each of dimension n − 1 : ( x + Δ x ) n = x n + n x n − 1 Δ x + ( n 2 ) x n − 2 ( Δ x ) 2 + ⋯ . {\displaystyle (x+\Delta x)^{n}=x^{n}+nx^{n-1}\Delta x+{\binom {n}{2}}x^{n-2}(\Delta x)^{2}+\cdots .} Substituting this into

4092-473: The n th derivative of a product of two functions in a form similar to that of the binomial theorem: ( f g ) ( n ) ( x ) = ∑ k = 0 n ( n k ) f ( n − k ) ( x ) g ( k ) ( x ) . {\displaystyle (fg)^{(n)}(x)=\sum _{k=0}^{n}{\binom {n}{k}}f^{(n-k)}(x)g^{(k)}(x).} Here,

4185-464: The recurrence relation The binomial coefficients occur in many areas of mathematics, and especially in combinatorics . In combinatorics the symbol ( n k ) {\displaystyle {\tbinom {n}{k}}} is usually read as " n choose k " because there are ( n k ) {\displaystyle {\tbinom {n}{k}}} ways to choose an (unordered) subset of k elements from

4278-568: The 13th century mathematical works of Yang Hui and also Chu Shih-Chieh . Yang Hui attributes the method to a much earlier 11th century text of Jia Xian , although those writings are now also lost. In 1544, Michael Stifel introduced the term "binomial coefficient" and showed how to use them to express ( 1 + x ) n {\displaystyle (1+x)^{n}} in terms of ( 1 + x ) n − 1 {\displaystyle (1+x)^{n-1}} , via "Pascal's triangle". Blaise Pascal studied

4371-463: The Indian mathematicians probably knew how to express this as a quotient n ! ( n − k ) ! k ! {\textstyle {\frac {n!}{(n-k)!k!}}} , and a clear statement of this rule can be found in the 12th century text Lilavati by Bhaskara . The first known formulation of the binomial theorem and the table of binomial coefficients appears in

4464-463: The answer is given by a binomial coefficient expression), for instance the number of words formed of n bits (digits 0 or 1) whose sum is k is given by ( n k ) {\displaystyle {\tbinom {n}{k}}} , while the number of ways to write k = a 1 + a 2 + ⋯ + a n {\displaystyle k=a_{1}+a_{2}+\cdots +a_{n}} where every

4557-440: The binomial coefficient ( n k ) {\displaystyle {\tbinom {n}{k}}} can be defined as the coefficient of the monomial X in the expansion of (1 + X ) . The same coefficient also occurs (if k ≤ n ) in the binomial formula (valid for any elements x , y of a commutative ring ), which explains the name "binomial coefficient". Another occurrence of this number

4650-1475: The binomial theorem this is equal to ( x 1 + y 1 ) n 1 ⋯ ( x d + y d ) n d = ∑ k 1 = 0 n 1 ⋯ ∑ k d = 0 n d ( n 1 k 1 ) x 1 k 1 y 1 n 1 − k 1 … ( n d k d ) x d k d y d n d − k d . {\displaystyle (x_{1}+y_{1})^{n_{1}}\dotsm (x_{d}+y_{d})^{n_{d}}=\sum _{k_{1}=0}^{n_{1}}\dotsm \sum _{k_{d}=0}^{n_{d}}{\binom {n_{1}}{k_{1}}}x_{1}^{k_{1}}y_{1}^{n_{1}-k_{1}}\dotsc {\binom {n_{d}}{k_{d}}}x_{d}^{k_{d}}y_{d}^{n_{d}-k_{d}}.} This may be written more concisely, by multi-index notation , as ( x + y ) α = ∑ ν ≤ α ( α ν ) x ν y α − ν . {\displaystyle (x+y)^{\alpha }=\sum _{\nu \leq \alpha }{\binom {\alpha }{\nu }}x^{\nu }y^{\alpha -\nu }.} The general Leibniz rule gives

4743-805: The binomial theorem, the expression on the right can be expanded, and then the real and imaginary parts can be taken to yield formulas for cos( nx ) and sin( nx ) . For example, since ( cos ⁡ x + i sin ⁡ x ) 2 = cos 2 ⁡ x + 2 i cos ⁡ x sin ⁡ x − sin 2 ⁡ x = ( cos 2 ⁡ x − sin 2 ⁡ x ) + i ( 2 cos ⁡ x sin ⁡ x ) , {\displaystyle \left(\cos x+i\sin x\right)^{2}=\cos ^{2}x+2i\cos x\sin x-\sin ^{2}x=(\cos ^{2}x-\sin ^{2}x)+i(2\cos x\sin x),} But De Moivre's formula identifies

SECTION 50

#1732780911891

4836-452: The binomials of the product. For example, there will only be one term x , corresponding to choosing x from each binomial. However, there will be several terms of the form x y , one for each way of choosing exactly two binomials to contribute a y . Therefore, after combining like terms , the coefficient of x y will be equal to the number of ways to choose exactly 2 elements from an n -element set. Expanding ( x + y ) yields

4929-414: The case where x and y are complex numbers. For this version, one should again assume | x | > | y | and define the powers of x + y and x using a holomorphic branch of log defined on an open disk of radius | x | centered at x . The generalized binomial theorem is valid also for elements x and y of a Banach algebra as long as xy = yx , and x

5022-438: The contributions to X in (1 + X ) (1 + X ) . As there is zero X or X in (1 + X ) , one might extend the definition beyond the above boundaries to include ( n k ) = 0 {\displaystyle {\tbinom {n}{k}}=0} when either k > n or k < 0 . This recursive formula then allows the construction of Pascal's triangle , surrounded by white spaces where

5115-980: The definition of binomial coefficients to be extended by replacing n by an arbitrary number α (negative, real, complex) or even an element of any commutative ring in which all positive integers are invertible: ( α k ) = α k _ k ! = α ( α − 1 ) ( α − 2 ) ⋯ ( α − k + 1 ) k ( k − 1 ) ( k − 2 ) ⋯ 1 for  k ∈ N  and arbitrary  α . {\displaystyle {\binom {\alpha }{k}}={\frac {\alpha ^{\underline {k}}}{k!}}={\frac {\alpha (\alpha -1)(\alpha -2)\cdots (\alpha -k+1)}{k(k-1)(k-2)\cdots 1}}\quad {\text{for }}k\in \mathbb {N} {\text{ and arbitrary }}\alpha .} With this definition one has

5208-508: The eponymous triangle comprehensively in his Traité du triangle arithmétique . However, the pattern of numbers was already known to the European mathematicians of the late Renaissance, including Stifel, Niccolò Fontana Tartaglia , and Simon Stevin . By the early 17th century, some specific cases of the generalized binomial theorem, such as for n = 1 2 {\displaystyle n={\frac {1}{2}}} , can be found in

5301-701: The equality holds for a given n ; we will prove it for n + 1 . For j , k ≥ 0 , let [ f ( x , y )] j , k denote the coefficient of x y in the polynomial f ( x , y ) . By the inductive hypothesis, ( x + y ) is a polynomial in x and y such that [( x + y )] j , k is ( n k ) {\displaystyle {\tbinom {n}{k}}} if j + k = n , and 0 otherwise. The identity ( x + y ) n + 1 = x ( x + y ) n + y ( x + y ) n {\displaystyle (x+y)^{n+1}=x(x+y)^{n}+y(x+y)^{n}} shows that ( x + y )

5394-950: The expansion of ( x + y ) on the right side in the n th row (numbered so that the top row is the 0th row): An example illustrating the last two points: ( x + y ) 3 = x x x + x x y + x y x + x y y + y x x + y x y + y y x + y y y ( 2 3  terms ) = x 3 + 3 x 2 y + 3 x y 2 + y 3 ( 3 + 1  terms ) {\displaystyle {\begin{aligned}(x+y)^{3}&=xxx+xxy+xyx+xyy+yxx+yxy+yyx+yyy&(2^{3}{\text{ terms}})\\&=x^{3}+3x^{2}y+3xy^{2}+y^{3}&(3+1{\text{ terms}})\end{aligned}}} with 1 + 3 + 3 + 1 = 2 3 {\displaystyle 1+3+3+1=2^{3}} . A simple example with

5487-515: The expression ( t k ) {\textstyle {\binom {t}{k}}} can be written as a polynomial with denominator k ! : this presents a polynomial in t with rational coefficients. As such, it can be evaluated at any real or complex number t to define binomial coefficients with such first arguments. These "generalized binomial coefficients" appear in Newton's generalized binomial theorem . For each k ,

5580-439: The factorial of n . This formula follows from the multiplicative formula above by multiplying numerator and denominator by ( n − k )! ; as a consequence it involves many factors common to numerator and denominator. It is less practical for explicit computation (in the case that k is small and n is large) unless common factors are first cancelled (in particular since factorial values grow very rapidly). The formula does exhibit

5673-2715: The first few cases of the binomial theorem: ( x + y ) 0 = 1 , ( x + y ) 1 = x + y , ( x + y ) 2 = x 2 + 2 x y + y 2 , ( x + y ) 3 = x 3 + 3 x 2 y + 3 x y 2 + y 3 , ( x + y ) 4 = x 4 + 4 x 3 y + 6 x 2 y 2 + 4 x y 3 + y 4 , ( x + y ) 5 = x 5 + 5 x 4 y + 10 x 3 y 2 + 10 x 2 y 3 + 5 x y 4 + y 5 , ( x + y ) 6 = x 6 + 6 x 5 y + 15 x 4 y 2 + 20 x 3 y 3 + 15 x 2 y 4 + 6 x y 5 + y 6 , ( x + y ) 7 = x 7 + 7 x 6 y + 21 x 5 y 2 + 35 x 4 y 3 + 35 x 3 y 4 + 21 x 2 y 5 + 7 x y 6 + y 7 , ( x + y ) 8 = x 8 + 8 x 7 y + 28 x 6 y 2 + 56 x 5 y 3 + 70 x 4 y 4 + 56 x 3 y 5 + 28 x 2 y 6 + 8 x y 7 + y 8 . {\displaystyle {\begin{aligned}(x+y)^{0}&=1,\\[8pt](x+y)^{1}&=x+y,\\[8pt](x+y)^{2}&=x^{2}+2xy+y^{2},\\[8pt](x+y)^{3}&=x^{3}+3x^{2}y+3xy^{2}+y^{3},\\[8pt](x+y)^{4}&=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4},\\[8pt](x+y)^{5}&=x^{5}+5x^{4}y+10x^{3}y^{2}+10x^{2}y^{3}+5xy^{4}+y^{5},\\[8pt](x+y)^{6}&=x^{6}+6x^{5}y+15x^{4}y^{2}+20x^{3}y^{3}+15x^{2}y^{4}+6xy^{5}+y^{6},\\[8pt](x+y)^{7}&=x^{7}+7x^{6}y+21x^{5}y^{2}+35x^{4}y^{3}+35x^{3}y^{4}+21x^{2}y^{5}+7xy^{6}+y^{7},\\[8pt](x+y)^{8}&=x^{8}+8x^{7}y+28x^{6}y^{2}+56x^{5}y^{3}+70x^{4}y^{4}+56x^{3}y^{5}+28x^{2}y^{6}+8xy^{7}+y^{8}.\end{aligned}}} In general, for

SECTION 60

#1732780911891

5766-407: The formula ( n k 1 , k 2 , … , k m ) = n ! k 1 ! ⋅ k 2 ! ⋯ k m ! . {\displaystyle {\binom {n}{k_{1},k_{2},\ldots ,k_{m}}}={\frac {n!}{k_{1}!\cdot k_{2}!\cdots k_{m}!}}.} Combinatorially,

5859-2778: The generalized binomial series gives the geometric series formula , valid for | x | < 1 : ( 1 + x ) − 1 = 1 1 + x = 1 − x + x 2 − x 3 + x 4 − x 5 + ⋯ . {\displaystyle (1+x)^{-1}={\frac {1}{1+x}}=1-x+x^{2}-x^{3}+x^{4}-x^{5}+\cdots .} More generally, with r = − s , we have for | x | < 1 : 1 ( 1 + x ) s = ∑ k = 0 ∞ ( − s k ) x k = ∑ k = 0 ∞ ( s + k − 1 k ) ( − 1 ) k x k . {\displaystyle {\frac {1}{(1+x)^{s}}}=\sum _{k=0}^{\infty }{-s \choose k}x^{k}=\sum _{k=0}^{\infty }{s+k-1 \choose k}(-1)^{k}x^{k}.} So, for instance, when s = 1/2 , 1 1 + x = 1 − 1 2 x + 3 8 x 2 − 5 16 x 3 + 35 128 x 4 − 63 256 x 5 + ⋯ . {\displaystyle {\frac {1}{\sqrt {1+x}}}=1-{\frac {1}{2}}x+{\frac {3}{8}}x^{2}-{\frac {5}{16}}x^{3}+{\frac {35}{128}}x^{4}-{\frac {63}{256}}x^{5}+\cdots .} Replacing x with -x yields: 1 ( 1 − x ) s = ∑ k = 0 ∞ ( s + k − 1 k ) ( − 1 ) k ( − x ) k = ∑ k = 0 ∞ ( s + k − 1 k ) x k . {\displaystyle {\frac {1}{(1-x)^{s}}}=\sum _{k=0}^{\infty }{s+k-1 \choose k}(-1)^{k}(-x)^{k}=\sum _{k=0}^{\infty }{s+k-1 \choose k}x^{k}.} So, for instance, when s = 1/2 , we have for | x | < 1 : 1 1 − x = 1 + 1 2 x + 3 8 x 2 + 5 16 x 3 + 35 128 x 4 + 63 256 x 5 + ⋯ . {\displaystyle {\frac {1}{\sqrt {1-x}}}=1+{\frac {1}{2}}x+{\frac {3}{8}}x^{2}+{\frac {5}{16}}x^{3}+{\frac {35}{128}}x^{4}+{\frac {63}{256}}x^{5}+\cdots .} The generalized binomial theorem can be extended to

5952-408: The infinitesimal change in the volume of an n -dimensional hypercube , ( x + Δ x ) n , {\displaystyle (x+\Delta x)^{n},} where the coefficient of the linear term (in Δ x {\displaystyle \Delta x} ) is n x n − 1 , {\displaystyle nx^{n-1},} the area of

6045-484: The left side sums the number of subsets of {1, ..., n } of sizes k = 0, 1, ..., n , giving the total number of subsets. (That is, the left side counts the power set of {1, ..., n }.) However, these subsets can also be generated by successively choosing or excluding each element 1, ..., n ; the n independent binary choices (bit-strings) allow a total of 2 n {\displaystyle 2^{n}} choices. The left and right sides are two ways to count

6138-659: The left side with ( cos ⁡ x + i sin ⁡ x ) 2 = cos ⁡ ( 2 x ) + i sin ⁡ ( 2 x ) {\displaystyle (\cos x+i\sin x)^{2}=\cos(2x)+i\sin(2x)} , so cos ⁡ ( 2 x ) = cos 2 ⁡ x − sin 2 ⁡ x and sin ⁡ ( 2 x ) = 2 cos ⁡ x sin ⁡ x , {\displaystyle \cos(2x)=\cos ^{2}x-\sin ^{2}x\quad {\text{and}}\quad \sin(2x)=2\cos x\sin x,} which are

6231-441: The multinomial coefficient ( n k 1 , ⋯ , k m ) {\displaystyle {\tbinom {n}{k_{1},\cdots ,k_{m}}}} counts the number of different ways to partition an n -element set into disjoint subsets of sizes k 1 , ..., k m . When working in more dimensions, it is often useful to deal with products of binomial expressions. By

6324-452: The multiplicative formula which using factorial notation can be compactly expressed as For example, the fourth power of 1 + x is and the binomial coefficient ( 4 2 ) = 4 × 3 2 × 1 = 4 ! 2 ! 2 ! = 6 {\displaystyle {\tbinom {4}{2}}={\tfrac {4\times 3}{2\times 1}}={\tfrac {4!}{2!2!}}=6}

6417-401: The next coefficient to its right until the k {\displaystyle k} -th one is reached. Due to the symmetry of the binomial coefficients with regard to k and n − k , calculation of the above product, as well as the recursive relation, may be optimised by setting its upper limit to the smaller of k and n − k . Finally, though computationally unsuitable, there

6510-485: The notation ( n k ) {\displaystyle {\tbinom {n}{k}}} in 1826, although the numbers were known centuries earlier (see Pascal's triangle ). In about 1150, the Indian mathematician Bhaskaracharya gave an exposition of binomial coefficients in his book Līlāvatī . Alternative notations include C ( n , k ) , n C k , C k , C n , C k , and C n , k , in all of which

6603-402: The number of different combinations (i.e. subsets) of b elements that can be chosen from an n -element set . Therefore ( n b ) {\displaystyle {\tbinom {n}{b}}} is usually pronounced as " n choose b ". Special cases of the binomial theorem were known since at least the 4th century BC when Greek mathematician Euclid mentioned

6696-503: The number of distinct sequences that define the same k -combination when order is disregarded. This formula can also be stated in a recursive form. Using the "C" notation from above, C n , k = C n , k − 1 ⋅ ( n − k + 1 ) / k {\displaystyle C_{n,k}=C_{n,k-1}\cdot (n-k+1)/k} , where C n , 0 = 1 {\displaystyle C_{n,0}=1} . It

6789-453: The number of ways of selecting k objects out of n without replacement, were of interest to ancient Indian mathematicians. The earliest known reference to this combinatorial problem is the Chandaḥśāstra by the Indian lyricist Pingala (c. 200 BC), which contains a method for its solution. The commentator Halayudha from the 10th century AD explains this method. By the 6th century AD,

6882-463: The numerator and denominator of the fraction . Although this formula involves a fraction, the binomial coefficient ( n k ) {\displaystyle {\tbinom {n}{k}}} is actually an integer . The binomial coefficient ( n k ) {\displaystyle {\tbinom {n}{k}}} can be interpreted as the number of ways to choose k elements from an n -element set. This

6975-426: The numerator of the first fraction, n k _ {\displaystyle n^{\underline {k}}} , is a falling factorial . This formula is easiest to understand for the combinatorial interpretation of binomial coefficients. The numerator gives the number of ways to select a sequence of k distinct objects, retaining the order of selection, from a set of n objects. The denominator counts

7068-1081: The ordinary derivative for c = 0 {\displaystyle c=0} , and the forward difference E − c − I {\displaystyle E^{-c}-I} for c < 0 {\displaystyle c<0} . The binomial theorem can be generalized to include powers of sums with more than two terms. The general version is ( x 1 + x 2 + ⋯ + x m ) n = ∑ k 1 + k 2 + ⋯ + k m = n ( n k 1 , k 2 , … , k m ) x 1 k 1 x 2 k 2 ⋯ x m k m , {\displaystyle (x_{1}+x_{2}+\cdots +x_{m})^{n}=\sum _{k_{1}+k_{2}+\cdots +k_{m}=n}{\binom {n}{k_{1},k_{2},\ldots ,k_{m}}}x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{m}^{k_{m}},} where

7161-599: The point is that with this definition all identities hold that one expects for exponentiation , notably ( 1 + X ) α ( 1 + X ) β = ( 1 + X ) α + β and ( ( 1 + X ) α ) β = ( 1 + X ) α β . {\displaystyle (1+X)^{\alpha }(1+X)^{\beta }=(1+X)^{\alpha +\beta }\quad {\text{and}}\quad ((1+X)^{\alpha })^{\beta }=(1+X)^{\alpha \beta }.} If α

7254-530: The polynomial ( t k ) {\displaystyle {\tbinom {t}{k}}} can be characterized as the unique degree k polynomial p ( t ) satisfying p (0) = p (1) = ⋯ = p ( k − 1) = 0 and p ( k ) = 1 . Its coefficients are expressible in terms of Stirling numbers of the first kind : The derivative of ( t k ) {\displaystyle {\tbinom {t}{k}}} can be calculated by logarithmic differentiation : This can cause

7347-401: The previous one by the symmetry of x and y in the first expression, and by comparison it follows that the sequence of binomial coefficients in the formula is symmetrical, ( n k ) = ( n n − k ) . {\textstyle {\binom {n}{k}}={\binom {n}{n-k}}.} A simple variant of the binomial formula

7440-776: The right hand side is ( n k ) + ( n k − 1 ) = ( n + 1 k ) , {\displaystyle {\binom {n}{k}}+{\binom {n}{k-1}}={\binom {n+1}{k}},} by Pascal's identity . On the other hand, if j + k ≠ n + 1 , then ( j – 1) + k ≠ n and j + ( k – 1) ≠ n , so we get 0 + 0 = 0 . Thus ( x + y ) n + 1 = ∑ k = 0 n + 1 ( n + 1 k ) x n + 1 − k y k , {\displaystyle (x+y)^{n+1}=\sum _{k=0}^{n+1}{\binom {n+1}{k}}x^{n+1-k}y^{k},} which

7533-420: The same collection of subsets, so they are equal. The formulas and follow from the binomial theorem after differentiating with respect to x (twice for the latter) and then substituting x = y = 1 . The Chu–Vandermonde identity , which holds for any complex values m and n and any non-negative integer k , is Chanda%E1%B8%A5%C5%9B%C4%81stra Too Many Requests If you report this error to

7626-424: The set {1, 2, 3, ..., n } and counting separately (a) the k -element groupings that include a particular set element, say " i ", in every group (since " i " is already chosen to fill one spot in every group, we need only choose k − 1 from the remaining n − 1 ) and (b) all the k -groupings that don't include " i "; this enumerates all the possible k -combinations of n elements. It also follows from tracing

7719-513: The special case of the binomial theorem for exponent n = 2 {\displaystyle n=2} . Greek mathematician Diophantus cubed various binomials, including x − 1 {\displaystyle x-1} . Indian mathematician Aryabhata 's method for finding cube roots, from around 510 CE, suggests that he knew the binomial formula for exponent n = 3 {\displaystyle n=3} . Binomial coefficients, as combinatorial quantities expressing

7812-1464: The sum of the 2 products of the form e 1 e 2 ... e n where each e i is x or  y . Rearranging factors shows that each product equals x y for some k between 0 and  n . For a given k , the following are proved equal in succession: This proves the binomial theorem. The coefficient of xy in ( x + y ) 3 = ( x + y ) ( x + y ) ( x + y ) = x x x + x x y + x y x + x y y _ + y x x + y x y _ + y y x _ + y y y = x 3 + 3 x 2 y + 3 x y 2 _ + y 3 {\displaystyle {\begin{aligned}(x+y)^{3}&=(x+y)(x+y)(x+y)\\&=xxx+xxy+xyx+{\underline {xyy}}+yxx+{\underline {yxy}}+{\underline {yyx}}+yyy\\&=x^{3}+3x^{2}y+{\underline {3xy^{2}}}+y^{3}\end{aligned}}} equals ( 3 2 ) = 3 {\displaystyle {\tbinom {3}{2}}=3} because there are three x , y strings of length 3 with exactly two y 's, namely, x y y , y x y , y y x , {\displaystyle xyy,\;yxy,\;yyx,} corresponding to

7905-483: The summation is taken over all sequences of nonnegative integer indices k 1 through k m such that the sum of all k i is  n . (For each term in the expansion, the exponents must add up to  n ). The coefficients ( n k 1 , ⋯ , k m ) {\displaystyle {\tbinom {n}{k_{1},\cdots ,k_{m}}}} are known as multinomial coefficients, and can be computed by

7998-422: The superscript ( n ) indicates the n th derivative of a function, f ( n ) ( x ) = d n d x n f ( x ) {\displaystyle f^{(n)}(x)={\tfrac {d^{n}}{dx^{n}}}f(x)} . If one sets f ( x ) = e and g ( x ) = e , cancelling the common factor of e from each term gives the ordinary binomial theorem. For

8091-513: The term X with an index i (running from 1 to n ), then each subset of k indices gives after expansion a contribution X , and the coefficient of that monomial in the result will be the number of such subsets. This shows in particular that ( n k ) {\displaystyle {\tbinom {n}{k}}} is a natural number for any natural numbers n and k . There are many other combinatorial interpretations of binomial coefficients (counting problems for which

8184-942: The theorem, the expansion of any nonnegative integer power n of the binomial x + y is a sum of the form ( x + y ) n = ( n 0 ) x n y 0 + ( n 1 ) x n − 1 y 1 + ( n 2 ) x n − 2 y 2 + ⋯ + ( n n − 1 ) x 1 y n − 1 + ( n n ) x 0 y n , {\displaystyle (x+y)^{n}={n \choose 0}x^{n}y^{0}+{n \choose 1}x^{n-1}y^{1}+{n \choose 2}x^{n-2}y^{2}+\cdots +{n \choose n-1}x^{1}y^{n-1}+{n \choose n}x^{0}y^{n},} where each ( n k ) {\displaystyle {\tbinom {n}{k}}}

8277-542: The three 2-element subsets of {1, 2, 3} , namely, { 2 , 3 } , { 1 , 3 } , { 1 , 2 } , {\displaystyle \{2,3\},\;\{1,3\},\;\{1,2\},} where each subset specifies the positions of the y in a corresponding string. Induction yields another proof of the binomial theorem. When n = 0 , both sides equal 1 , since x = 1 and ( 0 0 ) = 1. {\displaystyle {\tbinom {0}{0}}=1.} Now suppose that

8370-1968: The usual double-angle identities. Similarly, since ( cos ⁡ x + i sin ⁡ x ) 3 = cos 3 ⁡ x + 3 i cos 2 ⁡ x sin ⁡ x − 3 cos ⁡ x sin 2 ⁡ x − i sin 3 ⁡ x , {\displaystyle \left(\cos x+i\sin x\right)^{3}=\cos ^{3}x+3i\cos ^{2}x\sin x-3\cos x\sin ^{2}x-i\sin ^{3}x,} De Moivre's formula yields cos ⁡ ( 3 x ) = cos 3 ⁡ x − 3 cos ⁡ x sin 2 ⁡ x and sin ⁡ ( 3 x ) = 3 cos 2 ⁡ x sin ⁡ x − sin 3 ⁡ x . {\displaystyle \cos(3x)=\cos ^{3}x-3\cos x\sin ^{2}x\quad {\text{and}}\quad \sin(3x)=3\cos ^{2}x\sin x-\sin ^{3}x.} In general, cos ⁡ ( n x ) = ∑ k  even ( − 1 ) k / 2 ( n k ) cos n − k ⁡ x sin k ⁡ x {\displaystyle \cos(nx)=\sum _{k{\text{ even}}}(-1)^{k/2}{n \choose k}\cos ^{n-k}x\sin ^{k}x} and sin ⁡ ( n x ) = ∑ k  odd ( − 1 ) ( k − 1 ) / 2 ( n k ) cos n − k ⁡ x sin k ⁡ x . {\displaystyle \sin(nx)=\sum _{k{\text{ odd}}}(-1)^{(k-1)/2}{n \choose k}\cos ^{n-k}x\sin ^{k}x.} There are also similar formulas using Chebyshev polynomials . The number e

8463-513: The usual formula with factorials. However, for an arbitrary number r , one can define ( r k ) = r ( r − 1 ) ⋯ ( r − k + 1 ) k ! = ( r ) k k ! , {\displaystyle {r \choose k}={\frac {r(r-1)\cdots (r-k+1)}{k!}}={\frac {(r)_{k}}{k!}},} where ( ⋅ ) k {\displaystyle (\cdot )_{k}}

8556-447: The work of Henry Briggs ' Arithmetica Logarithmica (1624). Isaac Newton is generally credited with discovering the generalized binomial theorem, valid for any real exponent, in 1665, inspired by the work of John Wallis 's Arithmetic Infinitorum and his method of interpolation. A logarithmic version of the theorem for fractional exponents was discovered independently by James Gregory who wrote down his formula in 1670. According to

8649-793: The zeros, or the trivial coefficients, would be. A more efficient method to compute individual binomial coefficients is given by the formula ( n k ) = n k _ k ! = n ( n − 1 ) ( n − 2 ) ⋯ ( n − ( k − 1 ) ) k ( k − 1 ) ( k − 2 ) ⋯ 1 = ∏ i = 1 k n + 1 − i i , {\displaystyle {\binom {n}{k}}={\frac {n^{\underline {k}}}{k!}}={\frac {n(n-1)(n-2)\cdots (n-(k-1))}{k(k-1)(k-2)\cdots 1}}=\prod _{i=1}^{k}{\frac {n+1-i}{i}},} where

#890109