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MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 22, Friday, October 24

MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 22, Friday, October 24. 5.5. Binomial Identities. Homework (MATH 310#7F): Read Supplement (pp. 230-239) Do 5.5: All odd numbered exercises. Turn in 5.5: 10,12,20,26 Volunteers: ____________ ____________ Problem: 20.

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MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 22, Friday, October 24

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  1. MATH 310, FALL 2003(Combinatorial Problem Solving)Lecture 22, Friday, October 24

  2. 5.5. Binomial Identities • Homework (MATH 310#7F): • Read Supplement (pp. 230-239) • Do 5.5: All odd numbered exercises. • Turn in 5.5: 10,12,20,26 • Volunteers: • ____________ • ____________ • Problem: 20. • Bonus! Study 5.6 and implement any of the four algorithms. (up to 5 points/program).

  3. Binomial Coefficients - Revisited • C(n, r) = P(n, r)/P(r) = n!/(r!(n-r)!) • C(n, 0) = 1 • C(n, n) = 1 • C(n, r) = C(n-1, r) + C(n-1, r-1). • Combinatorial Proof of line 4.

  4. Pascal Triangle r = 2 n = 1 C(5,2)=10 n = 5

  5. Power of Combinatorics – The Birthday Paradox. • Do we have two people with the same birthday? • Let n be the number of persons. Let P(n) denote probability that all birthdays are distinct. • For n=2: • P(2) = 364/365. • For n=3: • P(3) = 364/365 363/365. • For general n: • P(n) = 365 364 ... (365-n+1)/365n.

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