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

Explore binomial coefficients and Pascal's Triangle applications in solving combinatorial problems. Study and implement algorithms for bonus points. Homework instructions included.

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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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