College Algebra Exam Review 48

College Algebra Exam Review 48 - 58 1. ALGEBRAIC THEMES...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 58 1. ALGEBRAIC THEMES Then arrange the first 2 (in 2Š D 2 ways) and the last n 2 (in .n 2/Š ways). This strange process for building permutations gives the formula Âà n nŠ D 2Š .n 2/Š : 2 Now dividing by 2Š .n 2/Š gives Âà nŠ n.n 1/ n D D : 2 2Š.n 2/Š 2 The virtue of this argument is that it remains valid if 2 is replaced by any k , 0 Ä k Ä n. So we have the following: Proposition 1.9.2. Let n be a natural number and let k be an integer in Âà n the range 0 Ä k Ä n. Let denote the number of k -element subsets of k an n-element set. Then Âà nŠ n : D k kŠ.n k /Š Âà n We extend the definition by declaring D 0 if k is negative or kÂà Âà 0 0 greater than n. Also, we declare D 1 and D 0 if k ¤ 0. Note 0 k à Âà Âà n n n that D D 1 for all n 0. The expression is generally read 0 n k as “n choose k .” Âà n Here are some elementary properties of the numbers . k Lemma 1.9.3. Let n be a natural number and k 2 Z. Âà n (a) is a nonnegative integer. kà   à n n (b) D . Âk à Ân kà  à n n1 n1 (c) D C . k k k1 Âà n Proof. Part (a) is evident from the definition of . k ...
View Full Document

This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

Ask a homework question - tutors are online