CSE 321 Discrete Structures

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CSE 321 Discrete Structures Winter 2008 Lecture 17 Counting TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA

Announcements Readings Counting Homework 5.3, (4.3) Permutations and Combinations 5.4, (4.4) Binomial Coefficients 5.5, (4.5) Generalized Permutations and Combinations Homework

Highlights from Lecture 16 Counting Product Rule |A1 A2| = |A1||A2| Sum Rule A1 and A2 disjoint, |A1 A2| = |A1| + |A2| Inclusion-Exclusion |A1 A2| = |A1| + |A2| - |A1 A2| Pigeon Hole Principle Example: How many license plates of the form LLLDDD that do not have 3 repeated letters or 3 repeated digits

Clever PHP Applications Every sequence of n2 + 1 distinct numbers contains a subsequence of length n+1 that is either strictly increasing or strictly decreasing. 4, 22, 8, 15, 19, 11, 2, 1, 9, 20, 10, 7, 16, 3, 6, 5, 14 Aside: design an efficient algorithm for finding the longest increasing subsequence

Proof Let a1, . . . am be a sequence of n2+1 distinct numbers Let ik be the length of the longest increasing sequence starting at ak Let dk be the length of the longest decreasing sequence starting at ak Suppose ik  n and dk  n for all k There must be k and j, k < j, with ik = ij and dk = dj

Permutations vs. Combinations How many ways are there of selecting 1st, 2nd, and 3rd place from a group of 10 sprinters? How many ways are there of selecting the top three finishers from a group of 10 sprinters?

r-Permutations An r-permutation is an ordered selection of r elements from a set P(n, r), number of r-permutations of an n element set

r-Combinations An r-combination is an unordered selection of r elements from a set (or just a subset of size r) C(r, n), number of r-permutations of an n element set

How many Binary strings of length 10 with 3 0’s How many different ways of assigning 38 students to the 5 seats in the front of the class How many different ways of assigning 38 students to a table that seats 5 students

Prove C(n, r) = C(n, n-r) [Proof 1] Proof by formula

Prove C(n, r) = C(n, n-r) [Proof 2] Combinatorial proof

Counting paths How many paths are there of length n+m-2 from the upper left corner to the lower right corner of an n  m grid?

Binomial Theorem

Binomial Coefficient Identities from the Binomial Theorem

Pascal’s Identity and Triangle