Combinatorial identity about binomial coefficients
In mathematics , Pascal's rule (or Pascal's formula ) is a combinatorial identity about binomial coefficients . It states that for positive natural numbers n and k ,
( n − 1 k ) + ( n − 1 k − 1 ) = ( n k ) , {\displaystyle {n-1 \choose k}+{n-1 \choose k-1}={n \choose k},} where
( n k ) {\displaystyle {\tbinom {n}{k}}} is a binomial coefficient; one interpretation of the coefficient of the
x k term in the
expansion of
(1 + x )n . There is no restriction on the relative sizes of
n and
k ,
[1] since, if
n < k the value of the binomial coefficient is zero and the identity remains valid.
Pascal's rule can also be viewed as a statement that the formula
( x + y ) ! x ! y ! = ( x + y x ) = ( x + y y ) {\displaystyle {\frac {(x+y)!}{x!y!}}={x+y \choose x}={x+y \choose y}} solves the linear two-dimensional difference equation
N x , y = N x − 1 , y + N x , y − 1 , N 0 , y = N x , 0 = 1 {\displaystyle N_{x,y}=N_{x-1,y}+N_{x,y-1},\quad N_{0,y}=N_{x,0}=1} over the natural numbers. Thus, Pascal's rule is also a statement about a formula for the numbers appearing in
Pascal's triangle .
Pascal's rule can also be generalized to apply to multinomial coefficients .
Combinatorial proof [ edit ] Illustrates combinatorial proof: ( 4 1 ) + ( 4 2 ) = ( 5 2 ) . {\displaystyle {\binom {4}{1}}+{\binom {4}{2}}={\binom {5}{2}}.} Pascal's rule has an intuitive combinatorial meaning, that is clearly expressed in this counting proof.[2] : 44
Proof . Recall that ( n k ) {\displaystyle {\tbinom {n}{k}}} equals the number of subsets with k elements from a set with n elements. Suppose one particular element is uniquely labeled X in a set with n elements.
To construct a subset of k elements containing X , include X and choose k − 1 elements from the remaining n − 1 elements in the set. There are ( n − 1 k − 1 ) {\displaystyle {\tbinom {n-1}{k-1}}} such subsets.
To construct a subset of k elements not containing X , choose k elements from the remaining n − 1 elements in the set. There are ( n − 1 k ) {\displaystyle {\tbinom {n-1}{k}}} such subsets.
Every subset of k elements either contains X or not. The total number of subsets with k elements in a set of n elements is the sum of the number of subsets containing X and the number of subsets that do not contain X , ( n − 1 k − 1 ) + ( n − 1 k ) {\displaystyle {\tbinom {n-1}{k-1}}+{\tbinom {n-1}{k}}} .
This equals ( n k ) {\displaystyle {\tbinom {n}{k}}} ; therefore, ( n k ) = ( n − 1 k − 1 ) + ( n − 1 k ) {\displaystyle {\tbinom {n}{k}}={\tbinom {n-1}{k-1}}+{\tbinom {n-1}{k}}} .
Algebraic proof [ edit ] Alternatively, the algebraic derivation of the binomial case follows.
( n − 1 k ) + ( n − 1 k − 1 ) = ( n − 1 ) ! k ! ( n − 1 − k ) ! + ( n − 1 ) ! ( k − 1 ) ! ( n − k ) ! = ( n − 1 ) ! [ n − k k ! ( n − k ) ! + k k ! ( n − k ) ! ] = ( n − 1 ) ! n k ! ( n − k ) ! = n ! k ! ( n − k ) ! = ( n k ) . {\displaystyle {\begin{aligned}{n-1 \choose k}+{n-1 \choose k-1}&={\frac {(n-1)!}{k!(n-1-k)!}}+{\frac {(n-1)!}{(k-1)!(n-k)!}}\\&=(n-1)!\left[{\frac {n-k}{k!(n-k)!}}+{\frac {k}{k!(n-k)!}}\right]\\&=(n-1)!{\frac {n}{k!(n-k)!}}\\&={\frac {n!}{k!(n-k)!}}\\&={\binom {n}{k}}.\end{aligned}}} Generalization [ edit ] Pascal's rule can be generalized to multinomial coefficients.[2] : 144 For any integer p such that p ≥ 2 {\displaystyle p\geq 2} , k 1 , k 2 , k 3 , … , k p ∈ N + , {\displaystyle k_{1},k_{2},k_{3},\dots ,k_{p}\in \mathbb {N} ^{+}\!,} and n = k 1 + k 2 + k 3 + ⋯ + k p ≥ 1 {\displaystyle n=k_{1}+k_{2}+k_{3}+\cdots +k_{p}\geq 1} ,
( n − 1 k 1 − 1 , k 2 , k 3 , … , k p ) + ( n − 1 k 1 , k 2 − 1 , k 3 , … , k p ) + ⋯ + ( n − 1 k 1 , k 2 , k 3 , … , k p − 1 ) = ( n k 1 , k 2 , k 3 , … , k p ) {\displaystyle {n-1 \choose k_{1}-1,k_{2},k_{3},\dots ,k_{p}}+{n-1 \choose k_{1},k_{2}-1,k_{3},\dots ,k_{p}}+\cdots +{n-1 \choose k_{1},k_{2},k_{3},\dots ,k_{p}-1}={n \choose k_{1},k_{2},k_{3},\dots ,k_{p}}} where
( n k 1 , k 2 , k 3 , … , k p ) {\displaystyle {n \choose k_{1},k_{2},k_{3},\dots ,k_{p}}} is the coefficient of the
x 1 k 1 x 2 k 2 ⋯ x p k p {\displaystyle x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{p}^{k_{p}}} term in the expansion of
( x 1 + x 2 + ⋯ + x p ) n {\displaystyle (x_{1}+x_{2}+\dots +x_{p})^{n}} .
The algebraic derivation for this general case is as follows.[2] : 144 Let p be an integer such that p ≥ 2 {\displaystyle p\geq 2} , k 1 , k 2 , k 3 , … , k p ∈ N + , {\displaystyle k_{1},k_{2},k_{3},\dots ,k_{p}\in \mathbb {N} ^{+}\!,} and n = k 1 + k 2 + k 3 + ⋯ + k p ≥ 1 {\displaystyle n=k_{1}+k_{2}+k_{3}+\cdots +k_{p}\geq 1} . Then
( n − 1 k 1 − 1 , k 2 , k 3 , … , k p ) + ( n − 1 k 1 , k 2 − 1 , k 3 , … , k p ) + ⋯ + ( n − 1 k 1 , k 2 , k 3 , … , k p − 1 ) = ( n − 1 ) ! ( k 1 − 1 ) ! k 2 ! k 3 ! ⋯ k p ! + ( n − 1 ) ! k 1 ! ( k 2 − 1 ) ! k 3 ! ⋯ k p ! + ⋯ + ( n − 1 ) ! k 1 ! k 2 ! k 3 ! ⋯ ( k p − 1 ) ! = k 1 ( n − 1 ) ! k 1 ! k 2 ! k 3 ! ⋯ k p ! + k 2 ( n − 1 ) ! k 1 ! k 2 ! k 3 ! ⋯ k p ! + ⋯ + k p ( n − 1 ) ! k 1 ! k 2 ! k 3 ! ⋯ k p ! = ( k 1 + k 2 + ⋯ + k p ) ( n − 1 ) ! k 1 ! k 2 ! k 3 ! ⋯ k p ! = n ( n − 1 ) ! k 1 ! k 2 ! k 3 ! ⋯ k p ! = n ! k 1 ! k 2 ! k 3 ! ⋯ k p ! = ( n k 1 , k 2 , k 3 , … , k p ) . {\displaystyle {\begin{aligned}&{}\quad {n-1 \choose k_{1}-1,k_{2},k_{3},\dots ,k_{p}}+{n-1 \choose k_{1},k_{2}-1,k_{3},\dots ,k_{p}}+\cdots +{n-1 \choose k_{1},k_{2},k_{3},\dots ,k_{p}-1}\\&={\frac {(n-1)!}{(k_{1}-1)!k_{2}!k_{3}!\cdots k_{p}!}}+{\frac {(n-1)!}{k_{1}!(k_{2}-1)!k_{3}!\cdots k_{p}!}}+\cdots +{\frac {(n-1)!}{k_{1}!k_{2}!k_{3}!\cdots (k_{p}-1)!}}\\&={\frac {k_{1}(n-1)!}{k_{1}!k_{2}!k_{3}!\cdots k_{p}!}}+{\frac {k_{2}(n-1)!}{k_{1}!k_{2}!k_{3}!\cdots k_{p}!}}+\cdots +{\frac {k_{p}(n-1)!}{k_{1}!k_{2}!k_{3}!\cdots k_{p}!}}={\frac {(k_{1}+k_{2}+\cdots +k_{p})(n-1)!}{k_{1}!k_{2}!k_{3}!\cdots k_{p}!}}\\&={\frac {n(n-1)!}{k_{1}!k_{2}!k_{3}!\cdots k_{p}!}}={\frac {n!}{k_{1}!k_{2}!k_{3}!\cdots k_{p}!}}={n \choose k_{1},k_{2},k_{3},\dots ,k_{p}}.\end{aligned}}} See also [ edit ] References [ edit ] ^ Mazur, David R. (2010), Combinatorics / A Guided Tour , Mathematical Association of America, p. 60, ISBN 978-0-88385-762-5 ^ a b c Brualdi, Richard A. (2010), Introductory Combinatorics (5th ed.), Prentice-Hall, ISBN 978-0-13-602040-0 Bibliography [ edit ] External links [ edit ] This article incorporates material from Pascal's triangle on PlanetMath , which is licensed under the Creative Commons Attribution/Share-Alike License .
This article incorporates material from Pascal's rule proof on PlanetMath , which is licensed under the Creative Commons Attribution/Share-Alike License .