Copyright © 2011 Pearson Education, Inc.

Combinations, Labeling, and the Binomial Theorem

Section 8.5

Sequences, Series,and Probability

Copyright © 2011 Pearson Education, Inc. Slide 8-3

8.5

Theorem: Combinations of n Things Taken r at a Time

The number of combinations of n things taken r at a time (or the number of subsets of size r from a set of n elements) is given by the formula

The notation or nCr may be used on your calculator

or elsewhere for C(n, r). Note that

.0!)!(

!),( nr

rrn

nrnC

for

.1!0!

!)0,(

!!0

!),(

n

nnC

n

nnnC and 1

r

n

Combinations of n ThingsTaken r at a Time

Copyright © 2011 Pearson Education, Inc. Slide 8-4

8.5

In a labeling problem, n distinct objects are to be given labels, each object getting exactly one label.

Labeling TheoremIf each of n distinct objects is to be assigned one label and there are r1 labels of the first type, r2 labels of the second type,…, and rk labels of the kth type, where r1 + r2 + ··· + rk = n, then the number of ways to assign the labels is

.!!!

!

21 krrr

n

Labeling

Copyright © 2011 Pearson Education, Inc. Slide 8-5

8.5

In general, the expansion of (a + b)n contains n + 1 terms in which the exponents have a sum of n.

The coefficient of an – rbr is the number of ways to label n factors with (n – r) of the a-labels and r of the b-labels:

The binomial theorem expresses this result using summation notation.

The Binomial TheoremIf n is a positive integer, then for any real numbers a and b,

)!(!

!

rnr

n

.)!(!

!,

0 rnr

n

r

nba

r

nba

n

r

rrnn

where

The Binomial Theorem