Combinations
Definition of Combination
An arrangement of objects in which
the order of selection does NOT
matter.
The keys to Combinations are:Similar to Permutations, there are no
repeats. But in contrast to permutations, order does not matter.
Ex: You have to visit three out of your four friends
houses: Andrew (A), Betty (B), Carlos (C), Dave (D).
What are the different ways to select the 3 houses to
visit?
A,B,C
A,B,D
A,C,D
B,C,D
Each arrangement is one combination of the elements A, B, C, and D.
In other words, there are 4 total combinations.
How to Calculate the Total Number of
Combinations
The total number of ways (without
repeats) to choose r objects from a set
of n objects (order does NOT matter).
Textbook
Definition
n!
OR
n Cr 
 n  r  !r !
Pr
r!
n
The method from the
last slide
Ex: Jim had 9 friends and needs to select 4
of them to go on a trip. How many
different arrangements are possible?
9
C4 
9!
 9  4 !4!

9!
5!4!

9876
4321
 126
Combination Example
A bag has 7 marbles (blue, green, red, yellow, orange,
purple, and black). If you select four marbles at once,
how many combinations are possible?
How many ways can you arrange 4 marbles from 7?
7
P4 
7!
 7  4!
 840
It is clear that selecting BGRY is not different than BRGY.
In the 840 permutations, how many times will B, G, R, Y be
repeated? How many ways can you arrange B, G, R, Y?
4! 24
If every combination is repeated 24 times, how many are
possible?
The # of Permutations
divided by the factorial of
the # of decisions.
840  24  35