Page 112 - Probability Demystified

CHAPTER 6 The Counting Rules 101

EXAMPLE: How many different signals can be made from seven different
flags if four flags are displayed in a row?

SOLUTION:

Hence n ¼ 7 and r ¼ 4, so
7! 7! 7 6 5 4 3!
P ¼ ¼ ¼ ¼ 7 6 5 4 ¼ 840
7 4
ð7 4Þ! 3! 3!

In the preceding examples, all the objects were different, but when some of
the objects are identical, the second permutation rule can be used.
The number of permutations of n objects when r 1 objects are identical,
r 2 objects are identical, etc. is
n!
r !r !. .. r !
p
2
1
where r 1 þ r 2 þ .. . þ r p ¼ n

EXAMPLE: How many different permutations can be made from the letters
of the word Mississippi?

SOLUTION:

There are 4 s, 4 i, 2 p, and 1 m; hence, n ¼ 11, r 1 ¼ 4, r 2 ¼ 4, r 3 ¼ 2, and r 4 ¼ 1
11! 11 10 9 8 7 6 5 4! 1,663,200
¼ ¼ ¼ 34,650
4! 4! 2! 1! 4! 4 3 2 1 2 1 1 48

EXAMPLE: An automobile dealer has 3 Fords, 2 Buicks, and 4 Dodges to
place in the front row of his car lot. In how many different ways by make of
car can he display the automobiles?

SOLUTION:

Let n ¼ 3 þ 2 þ 4 ¼ 9 automobiles; r 1 ¼ 3 Fords, r 2 ¼ 2 Buicks, and r 3 ¼ 4
Dodges; then there are 9! ¼ 9 8 7 6 5 4! ¼ 1260 ways to display the
3! 2! 4! 3 2 1 2 1 4!
automobiles.