College Algebra 9th txtbk

SECTION 8–4 Multiplication Principle, Permutations, and Combinations 537
But
We have developed Theorem 3.
n(n 1)(n 2) . . . (n r 1)(n r)! n!
Z THEOREM 3 Permutation of n Objects Taken r at a Time
The number of permutations of n objects taken r at a time is given by
or
P n,r n(n 1)(n 2) . . . (n r 1)
P n,r n!
(n r)!
y
r factors
0 r n
Note that if r n, then the number of permutations of n objects taken n at a time is
P n,n n!
(n n)! n!
0! n!
Recall, 0! 1.
which agrees with Theorem 2, as it should.
n
The permutation symbol P n,r also can be denoted by P r, n P r , or P(n, r). Many calculators
use n P r to denote the function that evaluates the permutation symbol.
EXAMPLE 5 Selecting Officers
From a committee of eight people, in how many ways can we choose a chair and a vicechair,
assuming one person cannot hold more than one position?
SOLUTION
We are actually asking for the number of permutations of eight objects taken two at a time—
that is, P 8,2 :
P 8,2 8!
(8 2)! 8!
6! 8 7 6! 56
6!
MATCHED PROBLEM 5
From a committee of ten people, in how many ways can we choose a chair, vice-chair, and
secretary, assuming one person cannot hold more than one position?
EXAMPLE 6 Evaluating P n,r
Find the number of permutations of 25 objects taken
(A) Two at a time
(B) Four at a time
(C) Eight at a time