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: 4]

Tyler needs to decide the order in which to schedule $11$ exams for his school. Two of these exams are Chemistry ($1$ SL and $1$ HL).

Find the number of different ways Tyler can schedule the $11$ exams given that the two Chemistry subjects must not be consecutive.

[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: 6]

A school basketball team of $5$ students is selected from $8$ boys and $4$ girls.

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

2. Determine how many teams can be formed consisting of $3$ boys and $2$ girls? [2]
   

3. Determine how many teams can be formed consisting of at most $3$ girls? [2]
   

[Maximum mark: 6]

In an art museum, there are 8 different paintings by Picasso, 5 different paintings by Van Gogh, and 3 different paintings by Rembrandt. The curator of the museum wants to hold an exhibition in a hall that can only display a maximum of 7 paintings at a time.

The curator wants to include at least two paintings from each artist in the exhibition.

1. Given that 7 paintings will be displayed, determine how many ways they can be selected. [4]
   

2. Find the probability that more Rembrandt paintings will be selected than Picasso paintings or Van Gogh paintings. [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: 6]

An arts and crafts store is offering a special package on personalized keychains.

The store has a selection of $6$ distinct types of charms.

Customers can personalize their keychains with up to $3$ distinct charms from the selection mentioned above.

Determine how many ways a customer can personalize a keychain if

1. The order of the selections is important. [3]
   

2. The order of the selections is not important. [3]
   

[Maximum mark: 6]

Ten students are to be arranged in a new chemistry lab. The chemistry lab is set out in two rows of five desks as shown in the following diagram.

1. Find the number of ways the ten students may be arranged in the lab. [1]
   

Two of the students, Hugo and Leo, were noticed to talk to each other during previous lab sessions.

2. Find the number of ways the students may be arranged if Hugo and Leo must sit so that one is directly behind the other. For example, Desk $1$ and Desk $6$. [2]
   

3. Find the number of ways the students may be arranged if Hugo and Leo must not sit next to each other in the same row. [3]
   

[Maximum mark: 7]

A professor and five of his students attend a talk given in a lecture series. They have a row of 8 seats to themselves.

Find the number of ways the professor and his students can sit if

1. the professor and his students sit together. [3]
   

2. the students decide to sit at least one seat apart from their professor. [4]
   

[Maximum mark: 6]

Julie works at a book store and has nine books to display on the main shelf of the store. Four of the books are non-fiction and five are fiction. Each book is different. Determine the number of possible ways Julie can line up the nine books on the main shelf, given that

1. the non-fiction books should stand together; [2]
   

2. the non-fiction books should stand together on either end; [2]
   

3. the non-fiction books should stand together and do not stand on either end. [2]
   

[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: 18]

Consider the family of polynomials of the form $P(x)=ax^3+bx^2+cx+d$ where coefficients $a$, $b$, $c$ and $d$ belong to the set $\{2,6,8,24\}$.

1. Find the number of possible polynomials if

   1. each coefficient value can be repeated;

   2. each coefficient must be different.[4]
      

Consider the case where $P(x)$ has $x+3$ as a factor, two purely imaginary roots, and all the coefficients are different.

2. 1. By considering the sum of the roots, find the two possible combinations for coefficients $a$ and $b$.

   2. Show that there is only one way to assign the values $a$, $b$, $c$, and $d$ if $P(0)=24$.[7]
      

Now, consider the polynomial with the coefficients found in part (b) (ii).

3. 1. Express $P(x)$ as a product of one linear and one quadratic factor.

   2. Determine the three roots of $P(x)$.[7]
      

[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]
   

[Maximum mark: 11]

Sophie and Ella play a game. They each have five cards showing roman numerals I, V, X, L, C. Sophie lays her cards face up on the table in order I, V, X, L, C as shown in the following diagram.

Ella shuffles her cards and lays them face down on the table. She then turns them over one by one to see if her card matches with Sophie's 4 card directly above. Sophie wins if no matches occur; otherwise Ella wins.

1. Show that the probability that Sophie wins the game is $\dfrac{11}{30}$.[6]
   

Sophie and Ella repeat their game so that they play a total of $90$ times. Let the discrete random variable $X$ represent the number of times Sophie wins.

2. Determine:

   1. the mean of $X$;

   2. the variance of $X$. [5]
      

[Maximum mark: 6]

The barcode strings of a new product are created from four letters A, B, C, D and ten digits $0,1,2,\dots,9$. No three of the letters may be written consecutively in a barcode string. There is no restriction on the order in which the numbers can be written.

Find the number of different barcode strings that can be created.

[Maximum mark: 25]

Jack and John have decided to play a game. They will be rolling a die seven times. One roll of a die is considered as one round of the game. On each round, John agrees to pay Jack $4 if $1$ or $2$ is rolled, Jack agrees to pay John $2 if $3,4,5$ or $6$ is rolled, and who is paid wins the round. In the end, who earns money wins the game.

1. Show that the probability that Jack wins exactly two rounds is $\dfrac{224}{729}$. [3]
   

2. 1. Explain why the total number of outcomes for the results of the seven rounds is $128$.

   2. Expand $(1 + y)^7$ and choose a suitable value of $y$ to prove that

      $$128 = \binom{7}{0} + \binom{7}{1} + \binom{7}{2} + \binom{7}{3} + \binom{7}{4} + \binom{7}{5} + \binom{7}{6} + \binom{7}{7}. \vspace{-0.5em}$$

   3. Give a meaning of the equality above in the context of the seven
      rounds.[4]
      

3. 1. Find the expected amount of money earned by each player in the game.

   2. Who is expected to win the game?

   3. Is this game fair? Justify your answer. [3]
      

4. Jack and John have decided to play the game again.

   1. Find an expression for the probability that John wins five rounds on the first game and two rounds on the second game. Give your answer in the form

      $$\binom{7}{r}^2\bigg[\frac{1}{3}\bigg]^s\bigg[\frac{2}{3}\bigg]^t \vspace{0.25em}$$

      where the values of $r,s$ and $t$ are to be found.

   2. Use your answer to (d) (i) and seven similar expressions to write down the probability that John wins a total of seven rounds over two games as the sum of eight probabilities.

   3. Hence prove that

      $$\binom{14}{7} = \sum_{k = 0}^7 \binom{7}{k}^2. \vspace{-0.5em}$$

      [9]
      

5. Now Jack and John roll a die $12$ times. Let $A$ denote the number of rounds Jack wins. The expected value of $A$ can be written as

   $$\mathrm{E}[A] = \sum_{r=0}^{12} r\binom{12}{r} \left[\dfrac{a^{12-r}}{b^{12}}\right] \vspace{-0.25em}$$

   1. Find the value of $a$ and $b$.

   2. Differentiate the expansion of $(1 + y)^{12}$ to prove that the expected
      number of rolls Jack wins is $4$. [6]