copyright © 2011 pearson education, inc. combinations, labeling, and the binomial theorem section...
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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