Combinations

A combination is a grouping of items in which order does not matter. There are generally fewer ways to select items when order does not matter. For example, there are 6 ways to order 3 items, but they are all the same combination: 6 permutations  {ABC, ACB, BAC, BCA, CAB, CBA} 1 combination  {ABC}

Combinations  There are also two types of combinations (remember the order does not matter now):  Repetition is Allowed: such as coins in your pocket (5,5,5,10,10)  No Repetition: such as lottery numbers (2,14,15,27,30,33)

Combinations with Repetition  These types of combinations are actually harder so we will not look at them

Combinations without Repetition  This is how lotteries work. The numbers are drawn one at a time, and if we have the lucky numbers (no matter what order) we win!lotteries  We already know that permutations tell us how many ways something can happen when order matters  But if order does not matter, the number of combinations will significantly reduce

Example  You go to an ice cream parlour and want to get 3 scoops of ice cream  The possible flavours are vanilla, chocolate, strawberry, banana, and lemon  How many possible combinations of three ice cream flavours are there assuming you only get one scoop of each flavor  Another way to word this would be to ask: “How many ways can I choose 3 ice cream flavours out of 5)

Example Cont  Let’s write it out S V CV B L S V BC B L S V LV C L S B CS B L S C LV C B Did we get them all??

How Can We Check? C(n, r) is similar to P(n, r). ‘n’ tells us how many things we have altogether. ‘r’ tells us how many we are choosing. This formula is very similar to our permutation formula, only we are dividing by r! r! is the number of ways our choices can be ordered. But order doesn’t affect combinations so we can divide it out. In our example, n = 5 and r = 3

Example  Evaluate ₉C₆

Example  Evaluate ₈C₄

Example  Evaluate ₂₁C₆

Application There are 12 different-colored cubes in a bag. How many ways can Randall draw a set of 4 cubes from the bag? Step 1 Determine whether the problem represents a permutation of combination. The order does not matter. The cubes may be drawn in any order. It is a combination.

Example Continued = 495 Divide out common factors. There are 495 ways to draw 4 cubes from Step 2 Use the formula for combinations. n = 12 and r = 4

Example  The swim team has 8 swimmers. Two swimmers will be selected to swim in the first heat. How many ways can the swimmers be selected?

Example  You are on your way to Hawaii (Aloha) and of 15 possible books your parents say you can only take 10. How many different collections of 10 books can you take?

Example  A committee of 5 members is to be selected from 6 seniors and 4 juniors. Find the number of ways in which this can be done if: a.there are no restrictions b.the committee has exactly 3 seniors c.the committee has at least 1 junior.

Example  A committee is to be chosen from a set of 7 women (including Mrs Smith) and 4 men (including Mr Smith). How many ways are there to form the committee if a) There are 5 committee members, 3 men and 2 women

A committee is to be chosen from a set of 7 women (including Mrs Smith) and 4 men (including Mr Smith). How many ways are there to form the committee if b) The committee can be any size except it must have equal numbers of men and women

A committee is to be chosen from a set of 7 women (including Mrs Smith) and 4 men (including Mr Smith). How many ways are there to form the committee if c) The committee has 4 people and one of them must be Mr Smith

A committee is to be chosen from a set of 7 women (including Mrs Smith) and 4 men (including Mr Smith). How many ways are there to form the committee if d) The committee has four people, two of each sex, and Mr and Mrs Smith cannot be on the same committee