Question

3. Give a combinatorial proof of the following identity. ("t?) = () + (-1) where n and k are positive integers with n > k.

Answer 1

3. ()+ (%) Samhet meneris ni (nokt) tulk k (nakt)! ina n! (no) le? (n-6 +1)! (nto! k (hak )! 2 ( )

Combinatorial Proof 1 Imagine that we are distributing n indistinguishable candies to k distinguishable children. By a direct application of Balls and Urns, there are (n+k-1 ways to do this. Alternatively, we can first give 0 <i<n candies to the oldest child so that we are essentially giving n - i I k-1 " Intk – 1) Intk - 2 – i) candies to k - 1 kids and again, with Balls and Urns, [ which simplifies to the desired result. k-1 Combinatorial Proof 2 k-2 In +1 We can form a committee of size k + 1 from a group of n + 1 people in ways. Now we hand out the numbers 1,2,3,...,n-k+1 to n - k + 1 of the n + 1 people. We can divide this into n-k+ 1 disjoint cases. In general, in cases, 1<x<n-k+1, person is on the committee and persons 1, 2, 3, ..., 3 - 1 are not on the committee. This can be done in (n-2+1 ways. Now we can sum the values of these n – k + 1 disjoint cases, getting (n+1) In-1 n - 2 kt k +1 k + +...+ k +1

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