We can use factorials to find all possible outcomes for multiple events that drop in value as items are selected.
Example
Consider the letters in the word JULY. In how many ways can you arrange all of the letters?
Solution
Super Subs makes three types of sandwiches: chicken salad, turkey or grilled cheese. Costumers then choose a side: chips, French Fries or fruit cup. And for a drink, they can have soda or water.
The tree diagram below shows all possible combinations.
Permutations
A permutation is an arrangement of objects in which order is important.
Example
Ten horses run in a race. In how many different ways can the horses finish first, second, and third?
(Assume there are no ties.)
Solution:
We only need to choose 3, first, second and third. So, 10 x 9 x 8 = 720
(10 choices for 1st) x (9 choices for 2nd) x (8 choices for 3rd) = 720
Example:
10P3 = 10 x 9 x 8 = 720
Consider the letters in the word
JULY.
In how many ways can you arrange 2 of the letters?
Combinations
A combination is a selection of objects in which order is not important.
Example 1:
In a drawing for 3 identical prizes, the order of the winners does not matter. If the prizes were different, then the order would matter. How many different ways can the identical prizes be awarded to 20 contestants?
Solution:
20C3 = 1,140
Example 2:
Count the possible combinations of 2 letters chosen from the list A, B, C, D.
Solution:
List all of the permutations of 2 letters from the list A, B, C, D. Because order is not important in a combination, cross out any duplicate pairs.
There are 6 possible combinations of 2 letters from the list A, B, C, D.
4C2 = 6
You order a sandwich at a restaurant. You can choose 2 side dishes from a list of 8. How many combinations of side dishes are possible?
Count the possible combinations of 4 letters chosen from the list P, Q, R, S, T, U.
Eight people serve on a committee. In how many different ways can a chairperson, a recorder, and a treasurer be chosen from the committee members?
You are listening to music. You have time to listen to 3 songs from your playlist of 16 songs. How many combinations of 3 songs are possible?
Count the possible combinations of 3 letters chosen from the list A, B, C, D, E.
