Probability and Statistics
Counting Methods Reference Sheet
Name: ______________________________
Block: ________
Counting
Method
Description
Formula
Fundamental
Counting
Principle
If one event can
occur in m ways
and a second
event can occur
in n ways, the
number of ways
the two events
can occur in
sequence is
𝑚 ∙ 𝑛.
𝑚∙𝑛
The number of
different ordered
arrangements of
n distinct objects
𝑛!
Permutations
of n objects
Example
You must choose a main food item (taco or burger), a side item (salad or
soup), and a dessert (ice cream, cake, or pie). How many distinct meals
are possible?
2 2 3 = 12
How many ways can you seat 5 people into 5 chairs?
5! = 5 4 3 2 1 = 120
Counting
Method
Description
Permutations
of n objects
taken r at a
time
The number of
different ordered
arrangements of
r distinct objects
selected from n
distinct objects,
where 𝑟 ≤ 𝑛
Formula
nPr
Example
𝑛!
= (𝑛−𝑟)!
How many different ways can you seat 12 people into 5 chairs?
12
The number of
distinguishable
ordered
arrangements of
Distinguishable n objects where
Permutations n1 objects are of
one type, n2
objects are of
another type, and
so on
Combinations
The number of
groups of r
objects selected
from a group of
n objects without
regard to order
𝑛!
𝑛1 ! ∙ 𝑛2 ! ∙∙∙ 𝑛𝑘 !
nCr
P5 =
12!
12! 12 11 10 9 8 7 6 5 4 3 2 1
=
=
= 12 11 10 9 8 = 95040
(12 − 5)! 7!
7 6 5 4 3 21
In how many distinguishable ways can you arrange the letters in the word
MISSISSIPPI?
11!
4! 4! 2! 1!
𝑛!
= (𝑛−𝑟)!∙𝑟!
How many committees of 4 people can you select from 10 people?
n!
10!
10!
=
=
= 210
n Cr =
(n − r )!r ! (10 − 4)!4! (6)!4!