Combinatorics

# Dierent mappings for this set thus the number of such

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Unformatted text preview:   When j=12, we have –  The coeﬃcient is CSCE 235 Combinatorics 45 Binomial Coeﬃcients (3) •  Many useful iden55es and facts come from the Binomial Theorem •  Corollary: Equali+es are based on (1+1)n=2n, ((- 1)+1)n=0n, (1+2)n=3n CSCE 235 Combinatorics 46 Binomial Coeﬃcients (4) •  Theorem: Vandermonde’s Iden5ty Let m,n,r be nonnega5ve integers with r not exceeding either m or n. Then •  Corollary: If n is a nonnega5ve integer then •  Corollary: Let n,r be nonnega5ve integers, r≤n, then CSCE 235 Combinatorics 47 Binomial Coeﬃcients: Pascal’s Iden5ty & Triangle •  The following is known as Pascal’s iden5ty which gives a useful iden5ty for eﬃciently compu5ng binomial coeﬃcients •  Theorem: Pascal’s Iden5ty Let n,k ∈Z+ with n≥k, then Pascal’s Iden5ty forms the basis of a geometric object known as Pascal’s Triangle CSCE 235 Combinatorics 4...
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