Permutations & Combinations

Probability

Warm-up

• How many distinguishable permutations are there for the letters in your last name?

Permutations

• An arrangement of a set of objects• Example: Most bankcards can be

used to access an account by entering a 4-digit PIN number. However knowing these 4 digits is not enough to access the account. The digits have to be in the correct order. Since the order is important, we must consider each arrangement as different

Combinations

• A selection from a group of objects without regard to order.

• If order were not important then any arrangement of the 4 numbers would access your bank account.

An expression for permutations• The number of permutations of n objects

taken r at a time reads “n permute r”

!!

rn

nPrn

Permutations with repetitions• The number of permutations of n

objects, where a are the same of one kind, b are the same of another kind, and c are the same of yet another kind, can be represented by the expression:

!!!

!

cba

n

Circular Permutations

• In general, the number of ways of arranging n objects around a circle is (n-1)!

• Example: At a graduation party, guests were seated in groups of 10 at circular tables. How many permutations are there for each table?

• (10-1)! = 9! = 362880

An expression for Combinations• The number of combinations of n

items taken r at a time reads:

“n choose r”

!!

!

rnr

nCrn

Diagonals in a Polygon

• How many diagonals are there in a octagon?

• Choose 2 points out of 8 to be joined, then don’t count adjacent pairs.

• 20

828 C

Question

• A group of 4 journalists is to be chosen to cover a murder trial. There are 5 male and 7 female journalists available. How many possible groups can be formed:

a) Consisting of 2 men and 2 women?

b) Consisting of a least one woman?

Solution a) 210 b) 490

Application to Probability

• Remember that

P(x) = # of favourable outcomes

total number of outcomes

• Two cards are picked without replacement from a deck of 52 cards. What is the probability that both are jacks?

Solution

P(2 jacks) = 2 jacks from 4 jacks

2 cards from 52 cards

= 4C2 = 6 = 1

52C2 1362 221

The student council forms a sub-committee of 5 council members to look at how funds raised should be spent. If there are a total of 15 student council members, 6 males and 9 females, what is the probability that the sub-committee will consist of exactly 4 females? At least 4 females?

• P(4 females, 1male) = (9C4)(6C1) = (126)(6) = 36

15C5 3003 143

• P(at least 4 females) = P(4 females) + P(5 females)

= (9C4)(6C1) + (9C5)(6C0) = 3

15C5 15C5 11