lecture30 - Lecture 30 Counting, Recurrence Relations...

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Lecture 30 Counting, Recurrence Relations
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Announcement Quiz on Friday Topics: Induction (Ch 4) and Counting (Ch 5)
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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
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Solution technique Imagine we have a cash box for seven types of bills with a compartment for each type. Represent this using 6 separators 100 | 50 | 20 | 10 | 5 | 2 | 1 For each bill place a * in the corresponding compartment That is: ** | | * | **| | | is the selection of two $100, one $20, two $10 bills
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Solution (cont) So the problem has been reduced to the number of ways of arranging six |s and five *s What’s the answer? C(6+5, 5)
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In general There are C(n+r-1, r) ways of choosing r objects with repetition from n classes (i.e. r-combinations with repetition from a set with n elements)
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Other Problems involving indistinguishable objects How many ways of arranging the letters S, U, C, C, E, S, S The answer is not (7!)
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Distinguishable Objects, Indistinguishable Boxes How many ways to assign four different objects to three identical offices (each office can accommodate up to four people)
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lecture30 - Lecture 30 Counting, Recurrence Relations...

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