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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 rpermutation of S. A
permutation of S is an npermutation.
The number of rpermutations 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 3permutations of S. Order matters here! 24 Number of rPermutations
Theorem: Let n and r be positive integers, r <= n.
Then P(n,r) = n(n1) ... (nr+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 n1 ways, ..., the rth element
can be chosen in (nr+1) ways. The claim follows
by the product rule. 25 Number of rPer...
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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.
 Fall '11
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