prob/stats definition a permutation is an ordered arrangement of objects. (for example, consider the...

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Prob/Stats Definition A permutation is an ordered arrangement of objects. (For example, consider the permutations of the letters A, B, C and D.)

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Page 1: Prob/Stats Definition A permutation is an ordered arrangement of objects. (For example, consider the permutations of the letters A, B, C and D.)

Prob/Stats

Definition

A permutation is an ordered arrangement of objects.

(For example, consider the permutations of the letters A, B, C and D.)

Page 2: Prob/Stats Definition A permutation is an ordered arrangement of objects. (For example, consider the permutations of the letters A, B, C and D.)

Prob/Stats

The number of permutations of nn objects is nn!

Page 3: Prob/Stats Definition A permutation is an ordered arrangement of objects. (For example, consider the permutations of the letters A, B, C and D.)

Prob/Stats

For example, consider how many permutations of the letters A, B, C and D we have, if we only select two at a time :

Permutations: {A, B} ; {A, C} ; {A, D} ; {B, A} ; {B, C} ; {B, D} ; {C, A} ; {C, B} ; {C, D} ; {D, A} ; {D, B} ; {D, C}

P(4, 2) = 12

Page 4: Prob/Stats Definition A permutation is an ordered arrangement of objects. (For example, consider the permutations of the letters A, B, C and D.)

Prob/Stats

PERMUTATIONS OF n OBJECTS TAKEN r AT A TIME:

P(n, r) = n!

(n – r )!

Page 5: Prob/Stats Definition A permutation is an ordered arrangement of objects. (For example, consider the permutations of the letters A, B, C and D.)

Prob/Stats

Example 1:

A club has 9 members. In how many ways can a president, vice-president, and secretary be chosen from the members of the club?

Answer: 504

Page 6: Prob/Stats Definition A permutation is an ordered arrangement of objects. (For example, consider the permutations of the letters A, B, C and D.)

Prob/Stats

Example 2:

A horse race has 10 horses entered. The 1st , 2nd, and 3rd place finishers will be announced. How many outcomes are there?

Answer: 720

Page 7: Prob/Stats Definition A permutation is an ordered arrangement of objects. (For example, consider the permutations of the letters A, B, C and D.)

Prob/Stats

Definition

A combination is an arrangement of objects where the order does not matter.

Page 8: Prob/Stats Definition A permutation is an ordered arrangement of objects. (For example, consider the permutations of the letters A, B, C and D.)

Prob/Stats

Again, consider how many combinations of the letters A, B, C and D we have, if we only select two at a time :

Combinations: {A, B} ; {A, C} ; {A, D} ; {B, C} ; {B, D} ; {C, D}

C(4, 2) = 6

Page 9: Prob/Stats Definition A permutation is an ordered arrangement of objects. (For example, consider the permutations of the letters A, B, C and D.)

Prob/Stats

COMBINATIONS OF n OBJECTS TAKEN r AT A TIME:

C(n, r ) = n!

r ! (n – r)!

Page 10: Prob/Stats Definition A permutation is an ordered arrangement of objects. (For example, consider the permutations of the letters A, B, C and D.)

Prob/Stats

Example 3:

A club has 9 members. How many ways can a committee of 3 be chosen from the members of the club?

Answer: 84

Page 11: Prob/Stats Definition A permutation is an ordered arrangement of objects. (For example, consider the permutations of the letters A, B, C and D.)

Prob/Stats

Example 4:

To win the UK National Lottery (in 1995) you had to correctly choose 6 different numbers, each being between 1 and 49. What is the probability of winning?

Answer: 1 in 13,983,816

Page 12: Prob/Stats Definition A permutation is an ordered arrangement of objects. (For example, consider the permutations of the letters A, B, C and D.)

Prob/Stats

Guidelines for Solving Counting Problems :

If the order matters, then use permutations.

If the order does not matter, use combinations.

Page 13: Prob/Stats Definition A permutation is an ordered arrangement of objects. (For example, consider the permutations of the letters A, B, C and D.)

Prob/Stats

Exercise 1: A poker hand consists of 5 cards randomly dealt from a standard 52-card deck. How many different poker hands are possible?

Answer: 2,598,960

Page 14: Prob/Stats Definition A permutation is an ordered arrangement of objects. (For example, consider the permutations of the letters A, B, C and D.)

Prob/Stats

Exercise 2: A hat contains 10 raffle tickets. Four tickets are to be selected. The holder of the 1st ticket wins a car, the 2nd wins a motorcycle, the 3rd wins a bicycle, and the 4th wins a wakeboard. How many different ways can these prizes be awarded?

Answer: 5040

Page 15: Prob/Stats Definition A permutation is an ordered arrangement of objects. (For example, consider the permutations of the letters A, B, C and D.)

Prob/Stats

Permutation OR Combination?

How many ways can you choose 4 books from a section of 7 different books?

How many ways can 4 books be arranged on a shelf from a selection of 7 different books?

How many batting lineups can you make for a 9-man baseball team?

How many basketball starting lineups can you create from a 12-man roster?

CombinationCombination

PermutationPermutation

PermutationPermutation

CombinationCombination

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