Combinations

 

Permutation  ARRANGEMENTS in which the order of the items are specified. Ex. Choosing a President and V.P. from a group of people  order matters when choosing In some cases, the order does not matter. Example. In a game of cards, what is in your hand is important, however the order in which it was dealt is not.

A selection from a group of items without regard to order is called a combination. How many ways can Chris, Mike, Leanne, Victoria, Don and Lucy be chosen for a committee that requires two people? 6 x 5 = 30 ways. It wasn’t important what order it was that the group needed to be in.

How many ways can these students fill the positions of President, Vice President and Secretary?  choosing three people out of a possible 6  6P3 = 120. The order matters. This is a permutation

How many ways can these people form a committee of 3 members? Number of combinations = = 6P3 / 3! = 120 / 6 = 20

Therefore if we look at the formula we can see that it is made up of nPr / r! nCr or C(n,r) or = =

Example How many different three topping pizzas can you get if there are 27 toppings to choose from? 27 toppings  n = 27. The pizza has three toppings, so r = 3. 27C3 = = =2925 possible pizzas

Example: A basketball team has 15 players on it. 20 players tried out for the team How many ways can the coach choose the players?

Use counting principles with combinations A flower company has on hand 12 kinds of rose bushes, 16 kinds of small shrubs, 11 kinds of evergreen seedling, and 18 kinds of flowering lilies. How many ways can the company fill an order if the customer wants 15 different varieties consisting of 4 roses, 3 shrubs, 2 evergreens, and 6 lilies? Roses  12C4 Shrubs  16C3 Evergreens  11C2 Lilies  18C6 Are these related to one another??? Yes! Therefore we need to multiply together

What if the customer wanted to have either 4 roses or 6 lilies? This is mutually exclusive  add together. Homework Pg 279 # 2ac, 3, 4, 5, 8, 11, 13, 22