Counting Principles

[Maximum mark: 4]

Find the number of ways in which twelve different baseball cards can be given to Emily, Harry, John and Olivia, if Emily is to receive $5$ cards, Harry is to receive $3$ cards, John is to receive $3$ cards and Olivia is to receive $1$ card.

[Maximum mark: 6]

A police department has $4$ male and $7$ female officers. A special group of $5$ officers is to be assembled for an undercover operation.

1. Determine how many possible groups can be chosen. [2]
   

2. Determine how many groups can be formed consisting of $2$ males and $3$ $\text{females.}$[2]
   

3. Determine how many groups can be formed consisting of at least one male. [2]
   

[Maximum mark: 4]

Peter needs to decide the order in which to schedule $14$ exams for his school. Two of these exams are Chemistry ($1$ SL and $1$ HL).

Find the number of different ways Peter can schedule the $14$ exams given that the two Chemistry subjects must not be consecutive.

[Maximum mark: 5]

Sophia and Zoe compete in a freestyle swimming race where there are no tied finishes and there is a total of $10$ competitors.

Find the total number of possible ways in which the ten swimmers can finish if Zoe finishes

1. in the position immediately after Sophia;[2]
   

2. in any position after Sophia.[3]
   

[Maximum mark: 6]

There are $11$ players on a football team who are asked to line up in one straight line for a team photo. Three of the team members named Adam, Brad and Chris refuse to stand next to each other. There is no restriction on the order in which the other team members position themselves.

Find the number of different orders in which the $11$ team members can be positioned for the photo.

[Maximum mark: 7]

There are six office cubicles arranged in a grid with two rows and three columns as shown in the following diagram. Aria, Bella, Charlotte, Danna, and Emma are to be stationed inside the cubicles to work on various company projects.

Find the number of ways of placing the team members in the cubicles in each of the following cases.

1. Each cubicle is large enough to contain the five team members, but Danna and Emma must not be placed in the same cubicle.[2]
   

2. Each cubicle may only contain one team member. But Aria and Bella must not be placed in cubicles which share a boundary, as they tend to get distracted by each other.[5]