Determining Sample Space Using Fundamental Counting Principle vs. Combinations
A. FUNDAMENTAL COUNTING PRINCIPLE: Use the fundamental counting principle to determine
the number of items in the sample space for the following events:
1a. Selecting a 4-digit pin number.
1b. Selecting a 4-digit pin number if the first digit can not be a zero.
1c. Selecting a 4-digit pin number if repetition of numbers is not allowed.
2. Selecting 3 officers (Pres, VP and Sec) from a club with 9 members.
3a. Generating a line up for the 9 players on a ball team.
3b. Generating a line up for the 9 players on a ball team if the pitcher bats first and the catcher bats
last.
4a. Putting 5 books in order on a shelf.
4b. Arranging the letters of the word "PRIME".
NOTE: Problems like number 3a, 4a, and 4b represent a calculation known as factorial.
Definition: n factorial = n! = (n)(n-1)(n-2)(n-3)…(1)
Example: 7! = (7)(6)(5)(4)(3)(2)(1) = 5040
B. PERMUTATIONS: When different orderings of the same items are counted separately, we have a
permutation problem:
n
Pr 
n!
(n  r )!
Example: The Board of Trustees at the author’s college has 9 members. Each year, they elect a
chairperson, vice chairperson, and secretary. How many different slates of candidates are possible?
C. COMBINATIONS: The fundamental counting principle, as demonstrated above, is used any time
the order of the outcomes is important. When selecting objects from a group where order is NOT
important, we use the formula for COMBINATIONS:
n
Cr  =
n!
r!(n  r )!
Example: From the debate team with a membership of 10 (n = 10) we are to select three members
to participate in the competition next week (r = 3). In how many ways can this task be accomplished?
Since selecting Sue, then Mary, then Ted would give the same results as Mary, then Ted, then Sue
(the competition team would still be the same) we use the combination formula:
n
Cr 10 C7 =
n!
10!
10  9  8  7  6  ... 10  9  8 720




 120
r!(n  r )! 3! (10  3)! 3  2  1  7  6  5... 3  2  1
6
Examples:
a. Calculate
8C3
b. There are 20 problems in the test bank and the instructor is selecting 5 problems to use on the
exam. How many different tests can be made?
c. In a team of 12 football players, three captains will be chosen to greet the opposing team. How
many ways can this be done?
d. Sally is packing to go on a trip. She has 7 dresses and will take 4 of them. How many different
ways can Sally fill the suitcase?
D. Summary: To determine the number of ways an experiment can be completed (the number of
items in the sample space):
When order makes a difference use the fundamental counting principle:
Lining up objects
Rearranging letters in a word with no repetition
Selecting outfits, food from a buffet, traveling from one place to another
Electing officers
When order does not make a difference use the formula for combinations nCr:
Selecting a small group from a large group
Awarding identical prizes
When order makes a difference but there is repetition among the items to select from use the
repetition formula:
Rearranging letters in a word with repetition among the letters
In each of the following problems, examine the experiment. Decide whether or not the order in
which the items are selected makes a difference. If it does, use the fundamental counting principle to
determine the number of items in the sample space. If it does not, use the formula for combinations.
If order does make a difference, but there are repeated items to select from, use the modified
permutation formula.
1. In how many different ways can a committee of three people be selected from a total of eight
people?
2. An ice cream parlor has 15 different flavors. George orders a sundae and has to select 3 flavors.
How many different selections are possible?
3. A teacher decides to give six identical prizes to 6 of the 30 students in the class. In how many
ways can the students be selected?
4. A teacher decides to give six different prizes to 6 of the 30 students in the class. In how many
ways can the students be selected?
5. A club consisting of eight people must choose a president, vice-president, and a secretary. How
many different arrangements are possible?
6. A student must select and answer four of five essay questions on a test. In how many ways can
he do this?
7. Chipola College must assign a new employee a 7 digit phone number, but the first three digits must
be 526. How many different numbers can be assigned if:
a) There are no restrictions on the remaining four numbers?
b) If the phone number can not have any digits repeated?
c) If the new employee is in the math dept and all math dept phone numbers end with 00?
8. Three of eight finalists will be selected and awarded $5000 scholarships. I how many way can this
be done?
9. The manager of a 9 member baseball team must arrange the batting order. In how many ways
can the manager of a baseball team arrange his players if:
a) There are no restrictions on the order?
b) The pitcher must bat last?
c) The catcher must bat first and the pitcher must bat last?
10. A man has 8 pairs of pants, 12 shirts, 15 ties, and 6 sport coats. If he wears one of each, how
many different outfits can he wear?
Worksheet answers: 1) 56 2) 455 3) 593,775 4) 427,518,000 5) 336 6) 5
7)a. 10,000 b. 840 c. 100 8) 56 9)a. 362,880 b. 40,320 c. 5,040 10) 8,640