MAT 2720 Discrete Mathematics Section 6 8 The

MAT 2720 Discrete Mathematics Section 6. 8 The Pigeonhole Principle http: //myhome. spu. edu/lauw

Goals l The Pigeonhole Principle (PHP) • First Form • Second Form

The Pigeonhole Principle (First Form) If n pigeons fly into k pigeonholes and k<n, some pigeonhole contains at least two pigeons.

Example 1 Prove that if five cards are chosen from an ordinary 52 - card deck, at least two cards are of the same suit.

Example 1 Prove that if five cards are chosen from an ordinary 52 - card deck, at least two cards are of the same suit.

Example 1 Prove that if five cards are chosen from an ordinary 52 - card deck, at least two cards are of the same suit. We can think of the 5 cards as 5 pigeons and the 4 suits as 4 pigeonholes. By the PHP, some suit (pigeonhole) is assigned to at least two cards (pigeons).

Example 1 l

First Form of PHP l Kind of very descriptive, not quantitative enough. . .

The Pigeonhole Principle (Second Form)

Example 1 (PHP 2 nd Form) Prove that if five cards are chosen from an ordinary 52 - card deck, at least two cards are of the same suit.

Example 1 (PHP 2 nd Form) l

Group Explorations l l l Partial steps are given. Think about how to adapt the 2 nd form of PHP to this new situation. Practice solving problem in a new situation. Keep your voices down. We will discuss it together at the end.

Group Explorations If 20 processors are interconnected, show that at least 2 processors are directly connected to the same number of processors.

Group Explorations