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). 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 1 11 121 1331 14641 1510 51 1615201561 n = 1 r = 2 n = 5 C(5,2)=10

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)/365 n. 1 1. 2 0.99726 3 0.991796 4 0.983644 5 0.972864 6 0.959538 7 0.943764 8 0.925665 9 0.905376 10 0.883052 11 0.858859 12 0.832975 13 0.80559 14 0.776897 15 0.747099 16 0.716396 17 0.684992 18 0.653089 19 0.620881 20 0.588562 21 0.556312 22 0.524305 23 0.492703 24 0.461656 25 0.4313 26 0.401759 27 0.373141 28 0.345539 29 0.319031 30 0.293684 31 0.269545 32 0.246652 33 0.225028 34 0.204683 35 0.185617 36 0.167818 37 0.151266 38 0.135932 39 0.12178 40 0.108768 41 0.0968484 42 0.0859695 43 0.0760771 44 0.0671146 45 0.0590241 46 0.0517472 47 0.0452256 48 0.039402 49 0.0342204 50 0.0296264