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

5.5. Binomial Identities Homework (MATH 310#7F): Read Supplement (pp ) 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). Bonus! Study 5.6 and implement any of the four algorithms. (up to 5 points/program).

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.

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

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. For general n: P(n) = (365-n+1)/365 n