Then r mn m n r rk k k0 in particular when choosing

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Unformatted text preview: he LHS, and the second way of counting (using the sum rule) yields the RHS of the following formula: ￿ n+1 k ￿ = ￿ ￿ ￿￿ n n + k−1 k 36 Vandermonde’s Identity Let m, n, and r be nonnegative integers, r ≤ min(n, m). Then ￿ ￿ ￿￿ ￿￿ ￿ r m+n m n = . r r−k k k=0 In particular, when choosing m = n = r, we get ￿ 2n n ￿ = n￿ ￿ k=0 37 n k ￿2 . Vandermonde’s Identity (2) Proof: We will prove this by counting in two different ways. Let S and T be two disjoint sets with m = |S | and and n = |T |. Counting in the first way: ￿n+m￿ We can choose r elements from S ∪ T in r ways. Counting in the second way: We can pick r elements from S ∪ T by picking r − k elements from S and k elements from T , where 0￿≤ k￿￿ ￿ . By the product ≤r m n rule, this can be done in r−k k ways. Hence the total number of ways to pick r elements from S ∪ T is ￿￿ ￿ r￿ ￿ m n . r−k k k=0 38 Permutations and Combinations with Repetitions 39 Motivation So far, we assumed that (a) the elements are clearly distinguishable and (b) each element is chosen at most once for a permutation and combinations. We will still keep the assumption (a) that the elements from which we choose are clearly distinguishable. However, we will now assume (b’) that each element can be chosen repeatedly in permutations and combinations. 40 Multisets A multiset is a generalization of the notion of a set in which elements are allowed to appear more than once. For example, {{a,a,b,c,c,d,d,d}} contains the elements - a with multiplicity 2, - b with multiplicity 1, - c with multiplicity 2, - d with multiplicity 3. 41 r-Combinations with Repetition Let S be a set with n elements. An r-combination with repetition of S is a multisubset with r elements of the set S . The number of r-combinations with repetitions of a set S with n elements is denoted by ￿￿ ￿￿ n . r 42 r-Combinations with Repetition Theorem. The number of r-combination with repetition of a set with n elements is given by ￿ ￿￿ ￿￿ ￿ n+r−1 n . = r r...
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