Combinations
Math I
Combinations


A selection of r objects from a group of n
objects where the order is not important.
The number of combinations of r objects
taken from a group of n distinct objects
is…
Example 1:

Eighteen basketball players are competing
for 5 starting positions. The players
selected to start will make up the first
team. If the order in which the players are
selected is not important, how many
different first teams are possible?
Solution:

The number of ways to choose 5 players
from 18 is:
18
!
18
!

18 C5 
((18  5)!5!) (13!5!)
 8568
On the Calculator:

TI–84 Plus:
18 MATH PRB (3) nCr 5 ENTER

8568
TI-30XIIS:
18 PRB nCr ENTER 5
8568
Example 2:

Your English teacher has asked you to
select 3 novels from list of 10 to read as
an independent project. In how many
ways can you choose which books to
read?
Solution:

Since you have 10
novels to choose from
and you are choosing
only 3…
10
C3  120
Deciding Whether to Add or
Multiply


Addition Principle:
“or”
If two or more events can not occur at the
same time, then you add the combinations
together
Multiplication Principle:
“and”
If two or more events occur at the same
time, then you multiply the combinations
together
Example 1 Part A:
1. A restaurant serves omelets that can be ordered with
any of the ingredients shown:


Vegetarian: green pepper, red pepper, onion,
mushroom, tomato, cheese
Meat: ham, bacon, sausage, steak
A. Suppose you want exactly 2 vegetarian ingredients and
1 meat ingredient in your omelet. How many different
types of omelets can you order?
Solution:


You can choose 2 of 6
vegetarian ingredients
and 1 of 4 meat
ingredients. So, the
number of possible
omelets is:
Since you are choosing
both for the same omelet,
you will multiply the
combinations together.
6
C 2  15
4
C1  4
15  4  60
Example 1 Part B:

B. Suppose you can afford at most 3
ingredients in your omelet. How many
different types of omelets can you order?
Solution:


You can order an omelet with
0, 1, 2, or 3 ingredients.
Because there are 10 items to
choose from…
10
You will either be choosing 0
OR 1 OR 2 OR 3 - you will add 10
the combinations.
C0 ,10 C1 ,10 C2 ,10 C3
C0 10 C1 10 C2 10 C3
 1 10  45 120  176
Example 2:

A movie rental business is having a special on
new releases. The new releases consist of 8
comedies, 3 family, 10 action, 7 dramas, and 2
mystery movies.
Part 1:
Suppose you want exactly 3 comedies and 2
dramas. How many different movie combinations
can you rent?
Solution:

You can choose 3 of the 8 comedies and 2 of the 7
dramas. So, the number of possible movie
combinations is:
8
C3
8!
8!


 56
(8  3)!3! 5!3!
7!
7 C2 
(7  2)!2!
7!

5!2!
56 x 21 = 1176
 21
Part 2:

Suppose you can afford at most 2 movies. How
many movie combinations can you rent?
Solution:
 You can rent 1 or 2 movies. Because there are
____
30 movies to choose from, the number of
movie combinations is…
Solution:

Since you can either
get 1 movie OR 2
movies…
C1
 30
C2
 435
30
30
30  435  465