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The Binomial Theorem: A Proof in One Step |
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The expression (x+y)^{n} is
This expression evaluates to the sum of terms involving x^{k}y^{n-k} for k = 0 to n. The number of times the term x^{k}y^{n-k} comes up is the number of ways of choosing x k times from the n binomials. Let that number be denoted as (^{n}_{k}). Therefore
Q.E.D.
The first x can be chosen from n different places. The second for (n-1) different places. The third from (n-2) different places. Thus the k-th x can be chosen from (n-(k-1)) different places. The total number is thus given by n(n-1)·…·(n-(k-1)). By the definition of the factorial function this is the same as n!/(n-k)!. It does not matter among the k choices of the places of x which one is first or second or so forth. This means the k! arrangements of the choices of x's are the same. Therefore the number of distinct ways of choosing x k times from the n binomials is
The most obvious corollary is that
Note that if x=1 and y=1 then the Binomial Theorem proves that
Likewise if x=−1 and y=1 then the Binomial Theorem proves that
Reference:
Arthur Benjamin and Jennifer J. Quinn, Proofs that Really Count: The Art of Combinatorial Proof, Mathematical Association of America, 2003.
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