1. Permutations & Combinations
MATH 102
Contemporary Math
S. Rook

2. Overview Section 13.3 in the textbook: Factorial notation Permutations
Combinations
Combining counting methods

3. Factorial Notation

4. Factorial Notation
Recall the problem of counting how many ways we can seat three men in three chairs
Because the product n x (n – 1) x … x 2 x 1 occurs often, we often write it in shorthand notation as n!
The exclamation point is pronounced factorial
n! means the product of n down to 1
3! = 3 · 2! = 3 · 2 · 1! = 3 · 2 · 1 = 6
1! AND 0! are both equivalent to 1
n! = n · (n – 1)!
We can expand a factorial into a product in order to quickly evaluate expressions containing factorials

5. Factorial Notation (Example)
Ex 1: Evaluate by hand: a) (8 – 5)! b) c)

6. Permutations

7. Permutations Recall that order affects a counting problem
Permutation means to count the number of possibilities when order among selections is important
e.g. From a room of four servants, how many ways can we select a group of three people if one is to cook, one is to chauffer, and one is to clean?
We can calculate a permutation using the Fundamental Counting Principle or:
Given a collection of n objects, the number of orderings of r of the objects is:

8. Permutations (Example)
Ex 2: On a biology quiz, a student must match eight terms with their definitions. Assume that the same term cannot be used twice. How many possibilities are there?

9. Permutations (Example)
Ex 3: How many ways can we select a president, vice-president, secretary and treasurer of an organization from a group of 10 people?

10. Combinations

11. Combinations
Combination means to count the number of possibilities when order among selections is NOT important
e.g. From a room of three servants, how many ways can we pick two valets for a party?
Consider starting with a permutation
Which choices list the same elements, but in a different order?
The number of possibilities for a combination must be smaller than the number for a permutation
Given a collection of n objects, the number of orderings of r of the objects

12. Combinations (Example)
Ex 4: Six players are to be selected from a 25-player Major League Baseball team to visit a school to support a summer reading program. How many different ways can the group of players be selected?

13. Combinations (Example)
Ex 5: Suppose a pizzeria offers a choice of 12 toppings. How many pizzas can be created with 4 toppings?

14. Combining Counting Methods

15. Combining Counting Methods
Recall that the F.C.P. considers counting problems occurring in stages
Sometimes the number of results of a stage can be a permutation or combination
Possible to have a mixture of permutations AND combinations in the same problem
It is ESSENTIAL to understand the difference between permutations and combinations!

16. Combining Counting Methods (Example)
Ex 6: Nicetown is forming a committee to investigate ways to improve public safety in the town. The committee will consist of three representatives from the seven-member town council, two members of a five-person citizens advisory board, and three of the 11 police officers on the force. How many ways can that committee be formed?

17. Combining Counting Methods (Example)
Ex 7: The students in the 12-member advanced communications design class at Center City Community College are submitting a project to a national competition. The must select a four-member team to attend the competition. The team must have a team leader and a main presenter while the other two equally-standing members have no particularly defined roles. In how many different ways can this team be formed?

18. Combining Counting Methods (Example)
Ex 8: Randy only likes movies and music. When writing a wishlist, he lists 10 movies and 6 music albums he would like to own. How many possibilities exist if his mother buys him 3 movies and 2 music albums from the wishlist?

19. Summary
After studying these slides, you should know how to do the following:
Evaluate expressions involving factorials
Differentiate between permutations & combinations
Apply permutations & combinations to solve counting problems
Additional Practice:
See problems in Section 13.3
Next Lesson:
The Basics of Probability Theory (Section 14.1)