section 8.6
Counting Principles
problem #1 - Eight pieces of paper are numbered from 1 to 8 and placed in a
box. One paper is drawn from the box at random, its number is
written down, and the paper is returned to the box. Then, a
second paper is drawn and its number written down. The two
numbers are added together. In how many different ways can
a sum of 12 be obtained?
problem #2 - Eight pieces of paper are numbered from 1 to 8 and placed in a
box. Two pieces of paper are randomly drawn, at the same
time, and their numbers are written down and totaled. In how
many different ways can a sum of 12 be obtained?
* In the previous two problems, it was relatively easy to list all of the ways in
which the events could occur. However, in many situations, there will be
too many ways to list; then, you can rely on some formulas and principles
such as the following.
Fundamental Counting Principle
Let E1 and E2 be two events. The first event E1 can occur in m different ways.
After E1 has occurred, E2 can occur in n different ways. The number of ways
that the two events can occur is m n .
problem #3 - How many different pairs of letters from the English alphabet
are possible?
problem #4 - Telephone numbers in the U.S. currently have 10 digits. The first
three are the area code and the next seven are the local
telephone number. How many different telephone numbers are
possible within each area code? (Note that a local telephone
number may not begin with either a "0" or a "1").
problem #5 - Three people, A, B, and C, comprise a club at school. In how
many different ways can a president and a treasurer be chosen?
In how many ways can a committee of two be formed?
permutation - a selection of elements in which no repetition occurs and the
order in which the elements occur is important
combination - a selection of elements in which no repetition occurs and the
order in which the elements occur is not important
problem #6 - Six horses are running a race. In how many different ways may
they finish the race? Is this a permutation or a combination?
Permutations of n elements, taken r at a time
The number of permutations of n elements, taken r at a time, is given by:
n Pr
=
n!
(n - r) !
Combinations of n elements, taken r at a time
The number of combinations of n elements, taken r at a time, is given by:
n Cr
=
n !
(n - r) ! r !
problem #7 - A baseball team has 15 players, all of whom are good enough
to play any position on the field. In how many different ways
can a starting infield be chosen (1st baseman, 2nd baseman,
shortstop, and 3rd baseman)?
* First solve the problem by hand, and then use the permutation
key on your calculator to verify your work.
Distinguishable Permutations
Suppose a set of n objects has n 1 of one kind, n 2 of a second kind, n 3 of a
third kind, and so on, with n = n1 + n + n 3 + . . . + n . Then, the number of
k
2
distinguishable permutations of the n objects is given by:
n!
n1 ! n ! n !
2
3
...
n !
k
* Continued on next page *
section 8.6 (continued)
problem #8 - In how many distinguishable ways can the letters in the word
BANANA be written?
problem #9 - A standard poker hand consists of 5 cards dealt from a
52-card deck. How many different poker hands are possible?
problem #10 - A 12-member swim team is being formed from 10 girls and
15 boys. The team must consist of five girls and seven boys.
How many different 12-member teams are possible?