Fundamental Counting Principle, Permutations and Combinations
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Fundamental Counting Principle, Permutations and Combinations
Fundamental Counting Principle
The number of ways in which a series of
successive things can occur is found by
multiplying the number of ways in which each
thing can occur.
Sally is packing for vacation. She wants to maximize her
wardrobe options. She plans on taking the following clothing
items. How many possible outfits will she have on her trip?
jeans 2 pairs, shorts – 3 pairs, skirts – 2, shirts – 9, shoes – 4 pairs
7 bottoms
Number of Outfits =
Example 1
252 7 • 9 • 4 =
Example 2:
You are taking a multiple choice test
that has five questions. Each of
the questions has five possible
answers with one correct answer
per question. If you select one
answer per question how many
ways can you answer the
questions?
Q1 Q2 Q3 Q4 Q5
5 • 5 • 5 • 5 • 5 = 55
Example 3
Telephone numbers in the U.S.
begin with a 3 digit area code
followed by a seven digit local
number. Area codes and local
numbers cannot begin with a 0 or
1. How many different telephone
numbers are possible?
Possible numbers per digit
8 • 10 • 10 • 8 • 10 • 10 • 10 • 10 • 10 • 10 = 6,400,000,000
area code local number
8 8 10 10 10 10 10 10 10 10
Permutations
An ordered arrangement that occurs when….
No item is used more than once.
The order of the arrangement makes a difference.
Example 4
You are the coach of a little league baseball team with 13 players.
You need to put together the batting order with nine players. How
many batting orders are possible?
Bat1 Bat 2 Bat 3 Bat 4 Bat 5 Bat 6 Bat 7 Bat 8 Bat 9
13
choices
11
choices
10
choices
9
choices
8
choices
7
choices
6
choices
5
choices
13 •12 • 11•10 • 9 • 8 • 7 • 6 • 5 = 259,459,200
12
choices
Permutations of n Things Taken r at a Time
The number of possible permutations if r items are taken
from n items is:
nPr = ______
n!
(n-r)!
Lets go back to our baseball batting order n = 13 r = 3
13P9 = _____ = __________________________ = 259, 459, 200
13! 13121110987654!
4! 4!
Example 5
A corporation has seven members on its board of
directors . In how many different ways can it select a
president, vice-president, secretary and treasurer?
840
Combinations
An arrangement that occurs when….
The items are selected from the same group.
No item is used more than once.
The order of the items makes no difference.
Distinguishing between Permutations & Combinations
Do you see the difference between a
permutation and a combination?
A permutation involves situations in which
order matters.
A combination involves situations in which
the order of the items makes no difference.
Example 6: Distinguishing between Permutations and Combinations
1. Six Students are running for student council president, vice-president
an treasurer. The student with the greatest number of votes becomes
the president, the student with the second greatest number of votes
becomes the vice-president and the student with the third greatest
number becomes the treasurer. How many different outcomes are
possible for these three positions?
Permutation or Combination
2. Six people are on the board of supervisors for the park district. A three
person committee is needed to study the possibility of expanding the
park. How many different committees could be formed from the 6
people?
Permutation or Combination
Example 6: Distinguishing between Permutations and Combinations
3. Baskin Robbins offers 31 different flavors of ice cream. One of their
items is a bowl consisting of three scoops of ice cream, each a
different flavor. How many such bowls are possible?
Permutation or Combination
4. In a race in which there are 50 runners and no ties, in how many
different ways can the first three finishers come in?
Permutation or Combination
Combinations of n Things Taken r at a Time
The number of possible combinations if r items are taken
from n items is:
ncr = ______
n!
(n-r)!r!
Example 7
From a group of 10 physicians , how many
ways can 4 doctors be selected to attend a
conference on accupuncture?
210
Example 8:
A person playing poker is dealt a 5 card hand from a
standard deck of 52 cards. The order in which the 5
cards are dealt does not matter. How many
different 5 card poker hands are possible?
= 2,598,960
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