1. Permutations and Combinations
Skill 19

2. Objectives
Compute number of ordered arrangements of outcomes using permutations.
Compute number of (nonordered) groupings of outcomes using combinations.

3. What are Factorials?

4. Permutations
To find the number of ordered arrangements of n objects taken as an entire group. Use the permutation formula.

5. Example-Permutations Rule
Compute the number of possible seating arrangements for eight people in five chairs.
Solution:
In this case, consider a total of n = 8 different people, and arrange r = 5 of these people.

6. Example–Solution
Using the multiplication rule, we get the same results
Permutations rule has the advantage of using factorials.

7. Combinations
In our previous counting formula, take the order of the objects or people into account.
When trying to find the number of non-ordered arrangements of n objects taken as different groupings or combinations.

8. Example–Combinations
In your political science class, you are assigned to read any 4 books from a list of 10 books. How many different groups of 4 are available from the list of 10?
Solution:
In this case, use combinations, rather than permutations, of 10 books taken 4 at a time.
Using n = 10 and r = 4, we have
There are 210 different groups of 4 books that can be selected from the list of 10.

9. Example–Permutations & Combinations
How many different combinations of management can there be to fill the positions of president, vice-president, and treasurer of a tennis club, knowing there are 16 members of the club?
Solution:
In this case, use permutations, of the 16 members there are 3 positions. Each person can only have one position.
Using n = 16 and r = 3, we have
𝑃 16,3 = 16! 16−3 !
= 16! 13!
=3360
There are 3360 different arrangements for the three positions of the tennis club.

10. Example– Permutations & Combinations
Mrs. Allan has six water bottles in her desk. In how many different ways can she choose two water bottles?
Solution:
In this case, use combinations, of the 6 water bottles it doesn’t matter which of the 2 she picks.
Using n = 6 and r = 2, we have
𝐶 6,2 = 6! 2! 6−2 !
= 6! 2!4!
=60
There are 60 different arrangements for picking two water bottles out of the desk.

11. Example– Permutations & Combinations
There are seven colors of the rainbow. How many different ways are there to choose a group of three colors?
Solution:
In this case, use combinations, of the 7 colors it doesn’t matter which of the 3 are chosen.
Using n = 7and r = 3, we have
𝐶 7,3 = 7! 3! 7−3 !
= 7! 3!4!
=35
There are 35 different arrangements for picking three colors of the rainbow.

12. Example–Permutations & Combinations
John has ten baseball bats in his bag one for each member of his team. Nine players bat each time through the lineup, and each player uses a different bat. In how many different orders can the bats be picked each time through the lineup?
Solution:
In this case, use permutations, there are 10 bats and 9 positions in the lineup. Each person can only have one position.
Using n = 10 and r = 9, we have
𝑃 10,9 = 10! 10−9 !
= 10! 1!
=3,628,800
There are 3,628,800 different arrangements for the bats to be dispersed.

13. Trees and Counting Techniques

14. 19: Permutations and Combinations
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