2.4 Combinations practice 2

How many triangles can you make using 6 non collinear points on a plane?

Consider a standard deck of 52 cards. How many different 5 card hands can be formed containing exactly 3 of the kind?

From a group of 7 men and 6 women, five people are to be selected to form a commitee. How many ways can this be down if there are at least 3 men on the committee?

In a group of 6 boys and 4 girls, four children are to be selected. In how many differnet ways can they be selected such that at least one boy is selected?

(Calculator) Richard has a standard deck of playing cards. He wants to know how many different hands are possible with the following properties

(a) A straight (5 cards of sequential rank i.e. 2,3,4,5,6)

(b) A straight but all Kings are taken out of the deck

(Calculator) You are the current commissioner of the Taiwan Lottery and you are considering making a small change. The current game has the player select 6 numbers out of a pool of 38 and then select one number out of a poor of 8. In order the increase the jackpot, you are tasked with increasing to total number of possible outcomes. Consider the following options where only one number is changed:
(a) Select 7 numbers out of a pool of 38 and then select one number out of a pool 8
(b) Select 6 numbers out of a pool of 39 and then select one number out of a pool 8
(c) Select 6 numbers out of a pool of 38 and then select one number out of a pool 9
Which one of these options will result in the most outcomes?

Louis, Ryan, Wish, Max, Joseph, and Chris are going on a road trip to Kenting. Of the group, only Louis, Ryan and Wish have driver’s license and can drive. They plan on renting one minivan that can seat all of them or two cars (each can seat 4 people including the driver). How many different ways can the group arrange themselves for the trip? (Other than the driver, where each person sits does not matter)

Identify the scenario as a combination or permutation. If the scenario is combination, type “A”; if the scenario is permutation, type “B”. Otherwise, type “C”.
(a) How many ways can you choose a captain and assistant captain on an 18 player team?

(b) In gym class, you are playing “Follow the Leader”. How many ways can you select 3 leaders from a class of 20?

(c) I have a pen. I have an apple. I have a pen. I have a pineapple. How many unique arrangements of these four objects is possible, i.e. pen-pineapple-apple-pen? Treat the pens as identical.

(Calculator) How many license plates of 4 symbols (letters and digits) can be made using at least two numbers in each if the symbols cannot repeat?

Evaluate each expression.

(a) [math]C^8_3[/math]

(b) [math]P^9_4[/math]

(c) [math]C^8_2[/math]

How many ways can 6 people work at 3 different classrooms if 2 people are in each classroom and 2 individuals refuse to work together?

(Calculator) Committee of 4 is chosen from 7 men and 6 women. How many different committees can be chosen if :

(a) there are no restrictions

(b) there must be 2 of each gender

(c) at least one of each gender is needed?

(Calculator) In how many ways can 5 boys and 4 girls be arranged on a bench if

(a) The boys and girls are in separate groups? (So a girl would not sit between 2 boys.)

(b) Anne and Jim wish to sit together?