Lecture3

# Lecture3 - Combinations Problem A certain state has license...

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Combinations 9/29/2005

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Problem: A certain state has license plates showing three numbers and three letters. How many diﬀerent license plates are possible 1. if the numbers must come before the letters? 2. if there is no restriction on where the letters and numbers appear? 1
As I was going to St. Ives, I met a man with seven wives, Each wife had seven sacks, Each sack had seven cats, Each cat had seven kits. Kits, cats, sacks and wives, How many were going to St. Ives? 2

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Binomial Coeﬃcients Deﬁnition. The number of distinct subsets with j elements that can be chosen from a set with n elements is denoted by ( n j ) . The number ( n j ) is called a binomial coeﬃcient . 3
Recurrence Relation Theorem. For integers n and j , with 0 < j < n , the binomial coeﬃcients satisfy: ± n j = ± n - 1 j + ± n - 1 j - 1 . 4

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Pascal’s triangle n = 0 1 10 1 10 45 120 210 252 210 120 45 10 1 9 1 9 36 84 126 126 84 36 9 1 8 1 8 28 56 70 56 28 8 1 7 1 7 21 35 35 21 7 1 6 1 6 15 20 15 6 1 5 1 5 10 10 5 1 4 1 4 6 4 1 3 1 3 3 1 2 1 2 1 1 1 1 j = 0 1 2 3 4 5 6 7 8 9 10 5
Theorem. The binomial coeﬃcients are given by the formula ± n j = ( n ) j j !

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## This note was uploaded on 07/16/2010 for the course MATH 20 taught by Professor Ionescu during the Fall '05 term at Dartmouth.

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Lecture3 - Combinations Problem A certain state has license...

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