CombinatoricsEdexcel A-Level Further Maths: Subtopic test

1. 1
   A club has 12 members and needs to choose a committee from them.

   (a)

   In how many ways can a committee of 4 members, with no special roles, be chosen?

   [1 mark]

   • A$11\,880$
   • B$20\,736$
   • C$495$
   • D$48$

   (b)

   Instead, the committee has a chair, a secretary, a treasurer and a publicity officer, all different members. In how many ways can these four posts be filled?

   [1 mark]

   • A$495$
   • B$20\,736$
   • C$48$
   • D$11\,880$

   (c)

   The club president is one of the 12 members. Find the number of committees of 4 members, with no special roles, that include the president.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Four-digit PINs are made from the digits $0$ to $9$. A PIN may begin with $0$.

   (a)

   How many different PINs are possible?

   [1 mark]

   • A$10\,000$
   • B$5040$
   • C$210$
   • D$40$

   (b)

   How many PINs have four different digits?

   [1 mark]

   • A$10\,000$
   • B$5040$
   • C$210$
   • D$9000$

   (c)

   Find the number of PINs that contain the digit $7$ at least once.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A football squad has 21 players: 2 goalkeepers and 19 outfield players. A team of 11 players is to be selected.

   (a)

   Find the number of different teams, with positions ignored, that contain exactly one goalkeeper.

   [3 marks]

   (b)

   Find the number of different teams, with positions ignored, that contain at least one goalkeeper.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A club has a squad of 10 athletes: 6 boys and 4 girls.

   (a)

   A relay team of 4 runners is chosen, running legs 1 to 4 in order.
   (i) Find the number of different relay teams, counting the order of the legs, that can be chosen from the squad.
   (ii) Find the number of these relay teams that have exactly two girls.

   [6 marks]

   (b)

   The coach chooses a group of athletes for a training camp. A group may contain any number of athletes from 1 to 10.
   (i) Find the number of different groups that can be chosen.
   (ii) Find the number of these groups that contain at least 3 girls.

   [6 marks]

   Total for question 4: 12 marks

End of questions