The difference between
permutations and combinations
Name:________________
Recap
A lunch special includes one main item, one
side, and one drink.
How many different meals can you choose if
you pick one main item, one side, and one
drink?
4 x 3 x 3 =36
Permutations and Combinations
A permutation is an arrangement of
outcomes in which the order does matter.
A combination is a grouping of outcomes
in which the order does not matter.
Permutations and Combinations
• Note: In order to complete a question on
permutations and combinations always ask
yourself the following question:
• Does the order of the terms matter?
Permutations and Arrangements
• A permutation is an arrangement of outcomes
in which the order matters.
• Questions on permutations and arrangements
will usually ask you ‘how many different ways’
can you arrange something. These questions
are very specific and will give you a concrete
order in which to follow.
ABCDE
If I wanted to arrange these letters, how many ways
could I do it if there are to be no restrictions ?
5 x 4 x 3 x 2 x 1 = 120
5 options
for the 1st
letter
4 options
for the 2nd
letter
3 options
for the 3rd
letter
2 options
for the 4th
letter
1 option
for the 5th
letter
• For the more straightforward questions use
the box method as shown below:
• Question
• How many ways can we arrange the letters of
the word Ireland if it must begin with the
letter E?
Question
• In how many ways can the letters of the word marketing be
arranged if:
1. There are no restrictions
2. The 3 vowels must be together
• All 3 vowels are not together?
• Not together = 1- (that they are together)
Question
Four girls and 5 boys are to be lined up:
How many arrangements are possible if:
1. There are no restrictions
2. If 2 particular girls must be together
3. If 2 particular girls must not be together
Permutations and arrangements which involve
taking a certain number of objects at a time.
• For questions of this format we must utilise the following
formula
• In the formula n represents the original number of
arrangements and r tells us how many are taken at a time.
• Question: In how many ways can 7 books be arranged on a
shelf taking 4 at a time
Combinations
• A combination is a grouping of outcomes in
which the order does not matter.
• Questions on combinations will usually give
you a scenario where the words ‘choose’ or
‘select’ are used.
• When answering questions on combinations
we must use the nCr button on our calculators
Question
• In how many ways can a team of 5 players be
chosen from 9 players? (Note: we can see here
that the order does not matter)
• When we have 2 events in a combination question
simply do out each event individually and multiply
your answers together
• Question
Find the number of ways in which a panel of four men
and three women can be chosen from seven men and
five women.
Question
• In how may ways can a committee of 6 be formed
from 5 teachers and 8 students if there are to be
more teachers than students on each committee.
(Hint: There must be at least one student on each
committee )
• Note: It is good practice for these questions to list out the
possible outcomes in English
The difference between Permutations and
Combinations
• Here are a few examples of combinations (order doesn’t
matter) and permutations (order matters).
1. Combination: Picking a team of 3 people from a group of 10.
2. Permutation: Picking a President, VP and Waterboy from a
group of 10.
A. Combination: Choosing 3 desserts from a menu of 10.
C(10,3) = 120.
B. Permutation: Listing your 3 favourite desserts, in order, from
a menu of 10. P(10,3) = 720.
Distinguishing between Combinations and
Permutations
Tell whether the following situations involve
combinations or permutations. Then give the
number of possible outcomes.
Remember to ask yourself:
Does the order of the terms matter?
Question 1
An English test contains five different essay
questions labeled A, B, C, D, and E. You are
supposed to choose 2 to answer. How many
different ways are there to do this?
Question 3
A family of 3 plans to sit in the same row at a
movie theater. How many ways can the family
be seated in 3 seats?
Question 4
Ingrid is stringing 3 different types of beads on
a bracelet. How many ways can she string the
next three beads if they must include one bead
of each type?
Question 5
John wants to order a sandwich with two of the
following ingredients: mushroom, eggplant,
tomato, and avocado. How many different
sandwiches can John choose?
Question 6
A group of 8 swimmers are swimming in a race.
Prizes are given for first, second, and third place.
How many different outcomes can there be?
Question 7
How many different ways can 9 people line
up for a picture?
Question 8
Four people need to be selected from a class
of 15 to help clean up the campus. How many
different ways can the 4 people be chosen?
Question 9
Four people need to be selected from a class of
15 to help clean up the campus. How many
different ways can the 4 people be chosen, if
the only two girls refuse to help?
Question 10
A basketball team has 12 members who can
play any position. How many different ways
can the coach choose 5 starting players?
Question 11
A basketball team has 12 members who can
play any position. How many different ways
can the coach choose 5 starting players if the
captain MUST play the first half?
Question 12
When ordering a pizza, you can choose 2
toppings from the following: mushrooms,
olives, pepperoni, pineapple, and sausage.
How many different types of pizza can you
order?
Question 13
Nine people in a writing contest are
competing for first, second and third prize.
How many ways can the 3 people be
chosen?
Question 14
You are ordering a triple-scoop ice-cream
cone. There are 18 flavours to choose from
and you don’t care which flavor is on the
top, middle, or bottom. How many different
ways can you select a triple-scoop icecream cone?