Lecture on Permutation Combination

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Unformatted text preview: elements of a set is an unordered selection of r elements from the set. Thus, an r‐combination is simply a subset of the set with r elements. The number of r‐combinations of a set with n distinct elements is denoted by C(n, r). The notation is also used and is called a binomial coefficient. Example: Let S be the set {a, b, c, d}. Then {a, c, d} is a 3‐ combination from S. It is the same as {d, c, a} since the order listed does not matter. C(4,2) = 6 because the 2‐combinations of {a, b, c, d} are the six subsets {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, and {c, d}. 8 Combinations Theorem : The number of r‐combinations of a set with n elements, where n r P(n, r) = C(n,r) ∙ P(r,r). Therefore, 9 Combinations Example: How many poker hands of five cards can be dealt from a standard deck of 52 cards? Also, how many ways are there to select 47 cards from a deck of 52 cards? Solution: Since the order in which the cards are dealt does not matter, the number of five card hands is: The different ways to select 47 cards from 52 is 10 Combinations Corollary : Let n and r be nonnegative integers with r n Then C(n, r) = C(n, n r Hence, C(n, r) = C(n, n r 11 Combinatorial Proofs Definition : A combinatorial proof of an identity is a proof that uses one of the following methods. A double counting proof uses counting arguments to prove that both sides of an identity count the same objects, but in different ways. A bijective proof shows that there is a bijection between the sets of objects counted by the two sides of the identity. 12 Combinatorial Proofs Here are two combinatorial proofs that C(n, r) = C(n, n r when r and n are nonnegative integers with r n: Bijective Proof: Suppose that S is a set with n elements. The function that maps a subset A of S to is a bijection between the subsets of S with r elements and the subsets with n r elements. Since there is a bijection between the two sets, they must have the same number of elements. Double Counting Proof: By definition the number of subsets of S with r elements is C(n, r). Each subset A of S can also be described by specifying which elements a...
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This document was uploaded on 03/29/2014 for the course COT 3100h at University of Central Florida.

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