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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Presentation transcript:

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 nn The number of permutations of n objects is n!

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

Prob/Stats PERMUTATIONS OF n OBJECTS TAKEN r AT A TIME: P(n, r) = n! (n – r )!

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

Prob/Stats Example 2: A horse race has 10 horses entered. The 1 st, 2 nd, and 3 rd place finishers will be announced. How many outcomes are there? Answer: 720

Prob/Stats Definition A combination is an arrangement of objects where the order does not matter.

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

Prob/Stats COMBINATIONS OF n OBJECTS TAKEN r AT A TIME: C(n, r ) = n! r ! (n – r)!

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

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

Prob/Stats Guidelines for Solving Counting Problems : If the order matters, then use permutations. If the order does not matter, use combinations.

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

Prob/Stats Exercise 2: A hat contains 10 raffle tickets. Four tickets are to be selected. The holder of the 1 st ticket wins a car, the 2 nd wins a motorcycle, the 3 rd wins a bicycle, and the 4 th wins a wakeboard. How many different ways can these prizes be awarded? Answer: 5040

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? Combination Permutation Permutation Combination