College Algebra 9th txtbk

538 CHAPTER 8 SEQUENCES, INDUCTION, AND PROBABILITY
SOLUTION Figure 4 shows the solution on a calculator.
Z Figure 4
MATCHED PROBLEM 6
Find the number of permutations of 30 objects taken
(A) Two at a time (B) Four at a time (C) Six at a time
Z Counting Combinations
Now suppose that an art museum owns eight paintings by a given artist and another art
museum hopes to borrow three of these paintings for a special show. How many ways can
three paintings be selected for shipment out of the eight available? Here, the order of the
items selected doesn’t matter. What we are actually interested in is how many subsets of
three objects can be formed from a set of eight objects. We call such a subset a combination
of eight objects taken three at a time. The total number of combinations is denoted
by the symbol
C 8,3
or
a 8 3 b
To find the number of combinations of eight objects taken three at a time, C 8,3 , we make
use of the formula for P n,r and the multiplication principle. We know that the number of permutations
of eight objects taken three at a time is given by P 8,3 , and we have a formula for
computing this quantity. Now suppose we think of P 8,3 in terms of two operations:
O 1 :
N 1 :
O 2 :
N 2 :
Select a subset of three objects (paintings)
C 8,3 ways
Arrange the subset in a given order
3! ways
The combined operation, O 1 followed by O 2 , produces a permutation of eight objects taken
three at a time. So,
To find C 8,3 , we replace P 8,3 in the preceding equation with 8!(8 3)! and solve for C 8,3 :
8!
(8 3)! C 8,3 3!
C 8,3
P 8,3 C 8,3 3!
8!
3!(8 3)! 8 7 6 5!
3 2 1 5! 56
The museum can make 56 different selections of three paintings from the eight available.
A combination of a set of n objects taken r at a time is an r-element subset of the
n objects. Reasoning in the same way as in the example, the number of combinations of n