18 pigeonhole principle example 1 suppose that there

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Unformatted text preview: are fewer than N objects total, contradicting our assumption on the total number of objects. 18 Pigeonhole Principle: Example 1 Suppose that there are 122 students in a class X. Then there must exist a week during which 122/52 =3 students of class X have a birthday. 19 Pigeonhole Principle: Example 2 Ten points are given within a square of unit size. Then there are two points that are closer to each other than 0.48. Proof: Let us partition the square into nine squares of side length 1/3, see Figure (a) below. (a) (b) By the pigeonhole principle, one square must contain at least 2 points, see Figure (b). The distance of two points within a square of side length 1/3 is at most (2/9)1/2 by Pythagoras’ theorem. The claim follows, since (2/9)1/2 < 0.471405 < 0.48. 20 Counting in Two Different Ways Rule When two different formulas enumerate the same set, then they must be the same. [In other words, you count the elements of the set in two different ways.] 21 Double Counting: Example Take an array of (n+1) x (n+1) dots. Thus, it contains (n+1)2 dots. Counting the points on the main diagonal, the upper diagonals, and the lower diagonals, we get n n 2 (n + 1) = (n + 1) + ￿ i=1 i+ ￿ i i=1 =⇒ n(n + 1) = (n + 1)2 − (n + 1) = 2 n(n + 1) =⇒ = 2 n ￿ i=1 i n ￿ i=1 i Permutations and Combinations 23 Permutations Let S be a set with n elements. An ordered arrangement of r elements of S is called an r-permutation of S. A permutation of S is an n-permutation. The number of r-permutations of a set with n elements is denoted by P(n,r). Example: S = {1,2,3,4}. Then (2,4,3) and (4,3,2) are two distinct 3-permutations of S. Order matters here! 24 Number of r-Permutations Theorem: Let n and r be positive integers, r <= n. Then P(n,r) = n(n-1) ... (n-r+1). Proof: Let S be a set with n elements. The first element of the permutation can be chosen in n ways, the second in n-1 ways, ..., the r-th element can be chosen in (n-r+1) ways. The claim follows by the product rule. 25 Number of r-Per...
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This note was uploaded on 03/24/2014 for the course CSCE 222 taught by Professor Math during the Fall '11 term at Texas A&M.

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