CHS Statistics
3.7: Counting
Objective: To find the counts of various
combinations and permutations, as well
as their corresponding probabilities
Warm-Up
• Alfred is trying to find an outfit to wear to take Beatrice on
their first date to Burger King. How many different ways can
he make an outfit out of this following clothes:
Pants: Green, Baby Blue, Black, Grey
Shirt: Red, Pink, Plaid, Blue, Lime Green
Tie: Polka dot, Stripped
Fundamental Counting Principle
• For a sequence of two events in which the first event can
occur in m ways and the second event can occur in n ways,
the events together can occur a total of m ∙ n ways.
• Example: You are purchasing a new car. The possible
manufacturers, car sizes, and colors are listed.
• Manufacturer: Ford, GM, Honda
• Car Size: compact, midsize
• Color: White, red, black, and green
How many different ways can you select one
manufacturer, one car size, and one color?
Fundamental Counting Principle (cont.)
• Example: The access code for a garage door consists of
four digits. How many codes are possible if:
• Each digit can be used only once and not repeated?
• Each digit can be repeated?
• Each digit can be repeated but the first digit cannot be 8
or 9?
•
Factorial Rule
• Examples:
• How many ways can 5 people be seated on a bench?
• How many ways can a class of 50 be ranked by grades?
• To answer questions like these, we will use the factorial rule.
• Factorial Rule
• A collection of n different items can be arranged in order n!
different ways.
• n! = n x (n – 1) x (n – 2) x (n – 3) x …
• 5! =
• 9! =
• 2! =
Factorial Rule (cont.)
• Examples:
• How many ways can 5 people be seated on a bench?
• How many ways can a class of 50 be ranked by grades?
Permutations
• Example: Forty-three sprinters race in a 5K. How many ways
can they finish first, second, and third?
• Can we use the factorial rule? Why or why not?
Permutations (When all Items Are Different)
• Permutations: When r items are selected from n available
items (without replacement).
• Therefore, the order matters.
n!
n pr 
(n  r )!
• Calculate the following permutations:
10
p3
7
p2
p3
24 p3
15
Permutations (cont.)
• Example: Forty-three sprinters race in a 5K. How many ways
can they finish first, second, and third?
Distinguishable Permutations
• When some items are identical
n!
n1!n2 !n3!  nk !
• If there are n items, 𝑛1 = first set of repeats, 𝑛2 = second
set of repeats, etc.
• Example: How many ways can you order AAAABBC?
Distinguishable Permutations (cont.)
• Example: A building contractor is planning to develop a
subdivision. The subdivision is to consist of 6 one-story
houses, 4 two-story houses, and 2 split-level houses. In how
many distinguishable ways can the houses be arranged?
Combinations
• Example: You are picking 3 different flavors to put on your
banana split. You can choose from 25 different flavors. How
many ways can this be done?
• Does the order matter here?
Combinations
• Combination Rule:
• When order does not matter, and we want to calculate the
number of ways (combinations) r items can be selected
from n different items.
n!
n Cr 
(n  r )!r!
• RECAP: When different orderings of the same items are
counted separately, we have a permutation problem, but
when different orderings of the same items are not counted
separately, we have a combination problem.
Combinations (cont.)
• Calculate the following combinations:
16
C4
12
C2
C3
25 C5
10
• Example: You are picking 3 different flavors to put on your
banana split. You can choose from 25 different flavors. How
many ways can this be done?
• Example: You want to buy three different CDs from a
selection of 5 CDs. How many ways can you make your
selection?
Combinations (cont.)
• Example: A state’s department of transportation plans to
develop a new section of interstate highway and receives 16
bids. The state plans to hire four of the companies. How
many different ways can the companies be selected?
• Example: The manager of an accounting department want
to form a three-person advisory committee from the 20
employees in the department. In how many ways can the
manager form this committee?
Probability Using Permutation and Combination
• A student advisory board consists of 17 members. Three
members serve as the board’s chair, secretary, and
webmaster. What is the probability of selecting at random
the three members that will hold these positions?
• You have 11 letters consisting of one M, four I’s, four S’s,
and two P’s. If the letters are randomly arranged in order,
what is the probability that the arrangement spells the word
Mississippi?
•
Probability Using Permutation and Combination (cont.)
• Find the probability of being dealt five diamonds from a
standard deck of playing cards?
• A food manufacturer is analyzing a sample of 400 corn
kernels for the presence of a toxin. In this sample, three
kernels have dangerously high levels of the toxin. If four
kernels are randomly selected from the sample, what is the
probability that exactly one kernel contains a dangerously
high level of the toxin?
Probability Using Permutation and Combination (cont.)
• A jury consists of five men and seven women. Three are
selected at random for an interview. Find the probability
that all three are men?
Assignment
3.7 Practice