This preview shows page 1. Sign up to view the full content.
Unformatted text preview: re not in A, i.e., those which are in . Since the complement of a subset of S with r elements has n r elements, there are also C(n, n r subsets of S with r elements.
Example: How many ways are there to select five players from a 10‐member tennis team to make a trip to a match at another school.
Solution: By Theorem 2, the number of combinations is
Example: A group of 30 people have been trained as astronauts to go on the first mission to Mars. How many ways are there to select a crew of six people to go on this mission?
Solution: By Theorem 2, the number of possible crews is 14 Section 15 Powers of Binomial Expressions
Definition: A binomial expression is the sum of two terms, such as x + y. (More generally, these terms can be products of constants and variables.)
We can use counting principles to find the coefficients in the expansion of (x + y)n
where n is a positive integer. To illustrate this idea, we first look at the process of expanding (x + y)3.
(x + y) (x + y) (x + y) expands into a sum of terms that are the product of a term from each of the three sums.
Terms of the form x3, x2y, x y2, y3 arise. The question is what are the coefficients? To obtain x3 , an x must be chosen from each of the sums. There is only one way to do this. So, the coefficient of x3 is 1. To obtain x2y, an x must be chosen from two of the sums and a y from the other. There are ways to do this and so the coefficient of x2y is 3. To obtain xy2, an x must be chosen from one of the sums and a y from the other two . There are ways to do this and so the coefficient of xy2 is 3. To obtain y3 , a y must be chosen from each of the sums. There is only one way to do this. So, the coefficient of y3 is 1. We have used a counting argument to show that (x + y)3 = x3 + 3x2y + 3x y2 + y3 .
Next we present the binomial theorem gives the coefficients of the terms in the expansion of (x + y)n . 16 Binomial Theorem Binomial Theorem: Let x and y be variables, and n a nonnegative integer. Then: Proof: We use combinatorial reasoning . The terms in the expansion of (x + y)n are...
View Full Document
This document was uploaded on 03/29/2014 for the course COT 3100h at University of Central Florida.
- Spring '08