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lecture29

# lecture29 - Lecture 29 Counting Recurrence Relations...

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Lecture 29 Counting, Recurrence Relations

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Announcement Quiz on Friday Topics: Induction (Ch 4) and Counting (Ch 5)
The binomial theorem j j n n j n y x j n y x 0 ) (

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Pascal’s Identity k n k n k n 1 1 Let’s prove it combinatorially
Pascal’s triangle 0 1 2 3 4 5 6 7 8 n =

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Pascal’s triangle 0 1 2 3 4 5 6 7 8 n = 1 2 4 8 16 32 64 128 256 sum = = 2 n
Permutations and Combinations with repetition In the definition of (standard) permutations and combinations, each item can be used at most once. What happens when we relax this? For example, how many strings of length r can be formed from the English alphabet? Use product rule. Sometimes call “ permutations with repetition”

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Combinations with repetition A cash box contains \$1, \$2, \$5, \$10, \$20, \$50, and \$100 bills How many ways to select 5 bills if The order of the bills is not important The bills are indistinguishable There are > 5 bills of each type
Solution technique Imagine we have a cash box for seven

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lecture29 - Lecture 29 Counting Recurrence Relations...

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