Permutations & Combinations

MATH 102Contemporary Math

S. Rook

Overview

• Section 13.3 in the textbook:– Factorial notation– Permutations– Combinations– Combining counting methods

Factorial Notation

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 13! = 3 · 2! = 3 · 2 · 1! = 3 · 2 · 1 = 6

1! AND 0! are both equivalent to 1n! = n · (n – 1)!

• We can expand a factorial into a product in order to quickly evaluate expressions containing factorials

Factorial Notation (Example)

Ex 1: Evaluate by hand:

a) (8 – 5)!

b)

c)

!3 !7

!10

!6

!2!8

Permutations

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:

!!

rn

nPrn

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?

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?

Combinations

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 !!

!

rnr

nCrn

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?

Combinations (Example)

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

Combining Counting Methods

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!

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?

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?

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?

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)