1.1 Counting Principles
Student Learning Outcome: You will be able to calculate the number of possibilities of a
given situation without explicitly writing them all out.
There are 3 methods that we will use to calculate the number of possibilities:
1. Fundamental Counting Principle
2. Permutations
a. Linear
b. Repetitions
c. Circular
3. Combinations
1. Fundamental Counting Principle
If a series of n decisions must be made, and if the first decision can be made in x1 ways, the
second decision can be made in x2 ways, the third decision can be made in x3 way, and so on,
then the number of different ways in which all of the decisions can be made is x1*x2*x3*…..xn
Ex: Five people are playing musical chairs with 4 chairs that are in a straight line. How many
possible seating arrangements exist?
Sometimes restrictions exist. Do the restrictions first.
Ex: How many 4 digit numbers are multiples of 2?
2 restrictions:
#1 - First digit  0
#2 - Last digit = {0, 2, 4, 6, 8}
2. Permutations
a) Linear: When arranging in order, we often use permutations
Note that 3! = 3  2  1 = 6
0! = 1 (by definition)
Ex: Only 5 of 12 different books can be placed upon the shelf. How many different
arrangements can be made?
b) Repetitions: Sometimes items are repeated or are identical. This reduces the number of
n!
arrangements available.
where a, b, and c are repeats
a !b !c !
Ex: How many arrangements of 12 textbooks can be made if there are 3 identical math
textbooks, 2 identical French textbooks, and 4 identical Chemistry books?
Ex: In how many ways can all of the letters of the word MINIMUM be arranged if:
a) there are no restrictions?
b) The arrangement must start with MMM?
c) The arrangement must start with M?
d) The arrangement must start with I and end in I?
c) Circular and Ring Permutations: When arranging items in a circle, the number of
arrangements is reduced. The first item must be the “point of reference”.
Ex: In how many ways can 4 couples be seated in a circle?
Ex: In how many ways can 4 couples be seated in a circle if each man must sit opposite his
wife?
Rings, bracelets, necklaces, key chains, or items in a circle that can be flipped have the number
of arrangements reduced again by a factor of 2. Right side up will give one pattern, and flipped
over will give a second pattern, but the items will not have changed position.
Ex: A charm bracelet will have 8 charms placed upon it. How many arrangements are possible
if all the charms are different?
3. Combinations
We use combinations when the order does not matter.
Ex: There are 10 girls and 8 boys being considered for a committee of 5 people. How many
different committees could exist if:
a) There are no restrictions?
b) there has to be 3 girls and 2 boys on the committee?
c) Jack is on the committee but Rose is not?
Math B30 1.1 Counting Principles Exercises
1. In how many ways can all of the letters of the word MAYOR be arranged?
3. In how many ways can all of the letters of the word COOKBOOK be arranged if each
arrangement must begin with B?
5. How many license plates, each of which contains three letters followed by three digits, can be
formed if the letters A, E, I, O, and U cannot be used and if the first digit cannot be a 0? Assume
that repetition of letters and digits is allowed.
7. Three digit numbers are to be made using the digits {0, 2, 3, 4, 5, 6, 7}. If repetition of digits
is not allowed, how many 3-digit numbers can be made:
a) if there are no other restrictions?
b) that are odd?
c) that are multiples of 5?
9. In how many ways can 7 different keys be arranged on a circular key ring?
11. There are 10 teams in a soccer league. How many games have to be played so that each team
will have played each of the other teams exactly once?
13. In how many ways can 2 identical red beads, 7 identical green beads, and 2 identical yellow
beads be strung along in a straight line?
15. Six people are to be seated at a circular table. Person A must be seated between persons B
and C for security reasons. How many arrangements are possible?
17. A box contains 7 blue marbles, 6 white marbles, and 4 red marbles. A set of three marbles is
to be selected from the box. How many sets will consist of:
a) 3 blue marbles?
b) 2 white marbles and 1 red marble?
c) no blue marbles?
19. How many 5-letter arrangements consisting of 2 vowels and 3 consonants, can be formed
from the letters in the word TRIANGLES if no letter is repeated?
21. A man has 5 coins in his pocket – a penny, a nickel, a dime, a quarter, and a looney. If he
leaves a tip at his table for the waiter, how many different amounts could he leave?
Math B30 1.1 Counting Principles Answers
1. 5! =120
3.
5. 21*21*21*9*10*10=8 334 900
7. a) 6*6*5=180 b) 5*5*3=75 (First choose the last digit)
c) 5*5*1=25 end in 5 and 6*5*1=30 end in 0 for a total of 55
9.
11.
(It takes a set of 2 teams to make a game.)
13.
15. 1(2!)(3!) = 12 (Person A must be seated. Then B and C can be arranged about A in
2! ways. The remaining 3 people can be arranged in 3! ways.)
17. a)
19.
21.
b)
c)