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MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 34, Friday, November 21

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7.1. Recurrence Relation Models Homework (MATH 310#10W): Read 7.1 Do 6.5: all odd numbered problems Turn in 6.5: 2,4,6,8 Turn in 7.1: 2,4,6,12,16,18,26,48

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Test 1 & 2 - Statistics Here is a stem- and-leaf report on the total. Median: 187 (Bonus points are not counted.) 1963322 187 179 16766 15 14 130 12 116

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Test 2 - Statistics Here is a stem- and-leaf report on the test. Median: 94 There are is only one mark. 1000 9664440 8 73 69 588

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Test 2 – Problem #1 Median 0 123456 0xxxxxx xxxx -2 -3x -4 -5 -6 -7 -8x -9 -10

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Test 2 – Problem #2 Median -1 1234567 0xx xxxxxxx -2x -3xx -4 -5 -6 -7 -8 -9 -10

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Test 2 – Problem #3 Median -3 123456 0x -2xx -3xxx -4 -5xxx -6xxx -7 -8 -9 -10

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Test 2 – Problem #4 Median -2 123456 0xxxx x -2x -3x -4xxx -5x -6 -7 -8 -9 -10x

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Test 2 – Problem #4 Median -2 123456 0 -2 -3 -4 -5 -6 -7 -8 -9 -10

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Test 2 – Problem #4 Median -2 123456 0 -2 -3 -4 -5 -6 -7 -8 -9 -10

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Test 2 – Problem #5 Median -5 123456 0 -2x -3x -4x -5xxxxx -6xx -7xx -8 -9 -10

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Test 2 – Problem #6 Median -3 123456 0x xx -2xx -3xx -4x -5xx -6x -7 -8x -9 -10

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Test 2 – Problem #7 Median 0 12345678 0xxxxxxxx x -2 -3x -4 -5 -6 -7x -8 -9 -10x

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Test 2 – Problem #8 Median -2 123456 0xxx x -2xxxxx -3x -4x -5x -6 -7 -8 -9 -10

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Test 2 – Problem #9 Median -4 123456 0x x -2xx -3x -4x -5xx -6 -7x -8x -9xx -10

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Test 2 – Problem #10 Median -1 123456 0xxx xxx -2xx -3xx -4x -5 -6 -7 -8x -9 -10

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Test 2 – Problem #11 Median -7 123456 0xxx -2 -3 -4 -5 -6xx -7xx -8xx -9x -10xx

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Recurrence Relation a n = c 1 a n-1 + c 2 a n-2 +... c r a n-r. a n = ca n-1 + f(n). a n = a 0 a n-1 + a 1 a n-2 +... + a n-2 a 1 + a n-1 a 0. a n,m = a n-1,m + a n,m-1. Initial Conditions. a n = a n-1 + a n-2, a 0 = 2, a 1 = 3.

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Example 1: Arrangements Find a recurrence relation for the number of ways to arrange n objects in a row. a n = na n-1. a 0 = 1.

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Example 2: Climbing Stairs n – stairs to climb each step can cover either 1 step or 2 steps. Find a recurrence relation for a n, the number of ways to climb the stairs. a n = a n-1 +a n-2. a 0 = a 1 = 1.

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