Basic committee/group selection

1 The 40 members of a club include Ranuf and Saed. All 40 members will travel to a concert. 35 members will travel in a coach and the other 5 will travel in a car. Ranuf will be in the coach and Saed will be in the car. In how many ways can the members who will travel in the coach be chosen?

1 The 26 members of the local sports club include Mr and Mrs Khan and their son Abad. The club is holding a party to celebrate Abad's birthday, but there is only room for 20 people to attend. In how many ways can the 20 people be chosen from the 26 members of the club, given that Mr and Mrs Khan and Abad must be included?

6

1. A collection of 18 books contains one Harry Potter book. Linda is going to choose 6 of these books to take on holiday.
   1. In how many ways can she choose 6 books?
   2. How many of these choices will include the Harry Potter book?
2. In how many ways can 5 boys and 3 girls stand in a straight line
   1. if there are no restrictions,
   2. if the boys stand next to each other?

3

1. How many different teams of 7 people can be chosen, without regard to order, from a squad of 15 ?
2. The squad consists of 6 forwards and 9 defenders. How many different teams containing 3 forwards and 4 defenders can be chosen?

5 A rugby union team consists of 15 players made up of 8 forwards and 7 backs. A manager has to select his team from a squad of 12 forwards and 11 backs.

1. In how many ways can the manager select the forwards?
2. In how many ways can the manager select the team?

6 A band has a repertoire of 12 songs suitable for a live performance. From these songs, a selection of 7 has to be made.

1. Calculate the number of different selections that can be made.
2. Once the 7 songs have been selected, they have to be arranged in playing order. In how many ways can this be done?

1 A girl is choosing tracks from an album to play at her birthday party. The album has 8 tracks and she selects 4 of them.

1. In how many ways can she select the 4 tracks?
2. In how many different orders can she arrange the 4 tracks once she has chosen them?

11

1. Calculate \({ } ^ { 5 } \mathrm { C } _ { 3 }\).
2. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 + 2 x ) ^ { 5 }\).

8 {} 6

(continued)

\multirow[t]{24}{*}{ 6 6 }	(ontnued)

4 The table gives the pay-off matrix for a zero-sum game between two players, Rowan and Colin. \begin{table}[h] \end{table} Rowan makes a random choice between strategies \(P\) and \(Q\), choosing strategy \(P\) with probability \(p\) and strategy \(Q\) with probability \(1 - p\).

1. Write down and simplify an expression for the expected pay-off for Rowan when Colin chooses strategy \(X\).
2. Using graph paper, draw a graph to show Rowan's expected pay-off against \(p\) for each of Colin's choices of strategy.
3. Using your graph, find the optimal value of \(p\) for Rowan.
4. Rowan plays using the optimal value of \(p\). Explain why, in the long run, Colin cannot expect to win more than 0.25 per game.

2 As part of a team-building exercise the reprographics technicians (Team R) and the computer network support staff (Team C) take part in a paintballing game. The game ends when a total of 10 'hits' have been scored. Each team has to choose a player to be its captain. The number of 'hits' expected by Team R for each pair of captains is shown below.

1. Complete the last two columns of the table in the insert.
2. State the minimax value and write down the minimax route.
3. Draw the network represented by the table.

6 Rowena and Colin play a game in which each chooses a letter. The table shows how many points Rowena wins for each combination of letters. In each case the number of points that Colin wins is the negative of the entry in the table. Both Rowena and Colin are trying to win as many points as possible. Colin's letter \begin{table}[h] \end{table}

1. Write down Colin's play-safe strategy, showing your working. What is the maximum number of points that Colin can win if he plays safe?
2. Explain why Rowena would never choose the letter \(W\). Rowena uses random numbers to choose between her three remaining options, so that she chooses \(X , Y\) and \(Z\) with probabilities \(x , y\) and \(z\), respectively. Rowena then models the problem of which letter she should choose as the following LP. $$\begin{array} { c l } \text { Maximise } & M = m - 1 \\ \text { subject to } & m \leqslant 2 x + 6 y + z , \\ & m \leqslant 4 x + 2 y , \\ & m \leqslant 3 y + 2 z , \\ & m \leqslant 2 x + 2 z , \\ & x + y + z \leqslant 1 \\ \text { and } & m \geqslant 0 , x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{array}$$
3. Show how the expression \(2 x + 6 y + z\) was formed. The Simplex algorithm is used to solve the LP problem. The solution has \(x = 0.3 , y = 0.2\) and \(z = 0.5\).
4. Show that the optimal value of \(M\) is 0.6 . Colin then models the problem of which letter he should choose in a similar way. When Rowena plays using her optimal solution, from above, Colin finds that he should never choose the letter \(N\). Letting \(p , q\) and \(t\) denote the probabilities that he chooses \(P , Q\) and \(T\), respectively, Colin obtains the following equations. $$- 3 p + q - t = - 0.6 \quad - p - 2 q + t = - 0.6 \quad p - q - t = - 0.6 \quad p + q + t = 1$$
5. Explain how the equation \(- 3 p + q - t = - 0.6\) is obtained.
6. Use the third and fourth equations to find the value of \(p\). Hence find the values of \(q\) and \(t\).

3 The 'Rovers' and the 'Collies' are two teams of dog owners who compete in weekly dog shows. The top three dogs owned by members of the Rovers are Prince, Queenie and Rex. The top four dogs owned by the Collies are Woof, Xena, Yappie and Zulu. In a show the Rovers choose one of their dogs to compete against one of the dogs owned by the Collies. There are 10 points available in total. Each of the 10 points is awarded either to the dog owned by the Rovers or to the dog owned by the Collies. There are no tied points. At the end of the competition, 5 points are subtracted from the number of points won by each dog to give the score for that dog. The table shows the score for the dog owned by the Rovers for each combination of dogs.

1. Explain why calculating the score by subtracting 5 from the number of points for each dog makes this a zero-sum game.
2. If the Rovers choose Prince and the Collies choose Woof, what score does Woof get, and how many points do Prince and Woof each get in the competition?
3. Show that column \(W\) is dominated by one of the other columns, and state which column this is.
4. Delete the column for \(W\) and find the play-safe strategy for the Rovers and the play-safe strategy for the Collies on the table that remains. Queenie is ill one week, so the Rovers make a random choice between Prince and Rex, choosing Prince with probability \(p\) and Rex with probability \(1 - p\).
5. Write down and simplify an expression for the expected score for the Rovers when the Collies choose Xena. Write down and simplify the corresponding expressions for when the Collies choose Yappie and for when they choose Zulu.
6. Using graph paper, draw a graph to show the expected score for the Rovers against \(p\) for each of the choices that the Collies can make. Using your graph, find the optimal value of \(p\) for the Rovers. If Queenie had not been ill, the Rovers would have made a random choice between Prince, Queenie and Rex, choosing Prince with probability \(p _ { 1 }\), Queenie with probability \(p _ { 2 }\) and Rex with probability \(p _ { 3 }\). The problem of choosing the optimal values of \(p _ { 1 } , p _ { 2 }\) and \(p _ { 3 }\) can be formulated as the following linear programming problem: $$\begin{array} { l l } \operatorname { maximise } & M = m - 4 \\ \text { subject to } & m \leqslant 6 p _ { 1 } + 5 p _ { 2 } , \\ & m \leqslant 3 p _ { 1 } + p _ { 2 } + 5 p _ { 3 } , \\ & m \leqslant 7 p _ { 1 } + 3 p _ { 2 } + 4 p _ { 3 } , \\ & p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1 \\ \text { and } & p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , p _ { 3 } \geqslant 0 , m \geqslant 0 . \end{array}$$
7. Explain how the expressions \(6 p _ { 1 } + 5 p _ { 2 } , 3 p _ { 1 } + p _ { 2 } + 5 p _ { 3 }\) and \(7 p _ { 1 } + 3 p _ { 2 } + 4 p _ { 3 }\) were obtained. Also explain how the linear programming formulation tells you that \(M\) is a maximin solution. The Simplex algorithm is used to find the optimal values of the probabilities. The optimal value of \(p _ { 1 }\) is \(\frac { 5 } { 8 }\) and the optimal value of \(p _ { 2 }\) is 0 .
8. Calculate the optimal value of \(p _ { 3 }\) and the corresponding value of \(M\).

3 The 'Rovers' and the 'Collies' are two teams of dog owners who compete in weekly dog shows. The top three dogs owned by members of the Rovers are Prince, Queenie and Rex. The top four dogs owned by the Collies are Woof, Xena, Yappie and Zulu. In a show the Rovers choose one of their dogs to compete against one of the dogs owned by the Collies. There are 10 points available in total. Each of the 10 points is awarded either to the dog owned by the Rovers or to the dog owned by the Collies. There are no tied points. At the end of the competition, 5 points are subtracted from the number of points won by each dog to give the score for that dog. The table shows the score for the dog owned by the Rovers for each combination of dogs.

1. Explain why calculating the score by subtracting 5 from the number of points for each dog makes this a zero-sum game.
2. If the Rovers choose Prince and the Collies choose Woof, what score does Woof get, and how many points do Prince and Woof each get in the competition?
3. Show that column \(W\) is dominated by one of the other columns, and state which column this is.
4. Delete the column for \(W\) and find the play-safe strategy for the Rovers and the play-safe strategy for the Collies on the table that remains. Queenie is ill one week, so the Rovers make a random choice between Prince and Rex, choosing Prince with probability \(p\) and Rex with probability \(1 - p\).
5. Write down and simplify an expression for the expected score for the Rovers when the Collies choose Xena. Write down and simplify the corresponding expressions for when the Collies choose Yappie and for when they choose Zulu.
6. Using graph paper, draw a graph to show the expected score for the Rovers against \(p\) for each of the choices that the Collies can make. Using your graph, find the optimal value of \(p\) for the Rovers. If Queenie had not been ill, the Rovers would have made a random choice between Prince, Queenie and Rex, choosing Prince with probability \(p _ { 1 }\), Queenie with probability \(p _ { 2 }\) and Rex with probability \(p _ { 3 }\). The problem of choosing the optimal values of \(p _ { 1 } , p _ { 2 }\) and \(p _ { 3 }\) can be formulated as the following linear programming problem: $$\begin{array} { l l } \operatorname { maximise } & M = m - 4 \\ \text { subject to } & m \leqslant 6 p _ { 1 } + 5 p _ { 2 } , \\ & m \leqslant 3 p _ { 1 } + p _ { 2 } + 5 p _ { 3 } , \\ & m \leqslant 7 p _ { 1 } + 3 p _ { 2 } + 4 p _ { 3 } , \\ & p _ { 1 } + p _ { 2 } + p _ { 3 } \leqslant 1 \\ \text { and } & p _ { 1 } \geqslant 0 , p _ { 2 } \geqslant 0 , p _ { 3 } \geqslant 0 , m \geqslant 0 . \end{array}$$
7. Explain how the expressions \(6 p _ { 1 } + 5 p _ { 2 } , 3 p _ { 1 } + p _ { 2 } + 5 p _ { 3 }\) and \(7 p _ { 1 } + 3 p _ { 2 } + 4 p _ { 3 }\) were obtained. Also explain how the linear programming formulation tells you that \(M\) is a maximin solution. The Simplex algorithm is used to find the optimal values of the probabilities. The optimal value of \(p _ { 1 }\) is \(\frac { 5 } { 8 }\) and the optimal value of \(p _ { 2 }\) is 0 .
8. Calculate the optimal value of \(p _ { 3 }\) and the corresponding value of \(M\).

2 Jane has a house on a Mediterranean island. She spends eight weeks a year there, either visiting twice for four weeks each trip or four times for two weeks each trip. Jane is wondering whether it is best for her to fly out and rent a car, or to drive out.
Flights cost \(\pounds 500\) return and car rental costs \(\pounds 150\) per week.
Driving out costs \(\pounds 900\) for ferries, road tolls, fuel and overnight expenses.

1. Draw a decision tree to model this situation. Advise Jane on the cheapest option. As an alternative Jane considers buying a car to keep at the house. This is a long-term alternative, and she decides to cost it over 10 years. She has to cost the purchase of the car and her flights, and compare this with the other two options. In her costing exercise she decides that she will not be tied to two trips per year nor to four trips per year, but to model this as a random process in which she is equally likely to do either.
2. Draw a decision tree to model this situation. Advise Jane on how much she could spend on a car using the EMV criterion.
3. Explain what is meant by "the EMV criterion" and state an alternative approach.

7

1. List the 15 partitions of the set \(\{ \mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } \}\) in which A and E are in the same subset.
2. By considering the number of subsets in each of the partitions in part (a), or otherwise, explain why there are 8 partitions of the set \(\{ \mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } \}\) into two subsets with A and E in different subsets. Ali says that each of the 15 partitions from part (a) can be used to give two partitions in which A and E are in different subsets by moving E into a subset on its own or by moving E into another subset.
   [0pt]
3. 1. By considering the partition from part (a) into just one subset, show that Ali is wrong. [1]
   2. By considering a partition from part (a) into more than two subsets, show that Ali is wrong.

4 In one department of a firm, four employees are selected for promotion from a staff of eighteen.

1. In how many ways can four employees be selected? It is known that throughout the firm 5\% of those selected for promotion decline it.
2. If 100 employees are randomly selected for promotion in the firm, calculate the number expected to decline promotion.
3. If 20 employees are selected at random for promotion, use the binomial distribution to find the probability that fewer than five employees will decline promotion.

Find the value of \(\frac{100!}{98! \times 3!}\) Circle your answer. [1 mark] \(\frac{50}{147}\) \quad \(1650\) \quad \(3300\) \quad \(161700\)

There are 15 students enrolled in a Maths club.

1. In how many ways is it possible to choose 4 of the students to take part in a competition? [2]

There are 4 different medals to be allocated, at random, to the students in the Maths club.

1. If there are no restrictions about how many medals a student may receive, in how many ways can the medals be allocated? [2]
2. Find the probability that no student receives more than one medal. [3]