Mathematics 9709/52 — February/March 2020

Walkthrough

First identify which people are already assigned. Ranuf is in the coach and Saed is in the car, so when choosing the coach members, Ranuf is automatically included and Saed is automatically excluded. The coach needs 35 members, but one of those is already Ranuf, so we only need to choose 34 more. The car has 5 members, and Saed is one of them, so Saed cannot be one of the 34 chosen. This leaves $40 - 2 = 38$ people from whom to choose. Therefore the number of ways is the combination $^{38}C_{34}$.

This is a combination, not a permutation, because the order in which the coach members are chosen does not matter; we are only selecting a set of people to travel in the coach. Once the coach members are chosen, the remaining 5 people automatically travel in the car, but Saed is guaranteed to be among them because he was never eligible for the coach.

Key Takeaways

• Fixed members reduce the number of people available to choose from.
• A selection without regard to order uses combinations, not permutations.
• $^{n}C_{r} = ^{n}C_{n-r}$, so $^{38}C_{34}$ can be written as $^{38}C_{4}$.
• Always check which people are already placed before counting.

Common Mistakes

• Choosing 35 members from all 40 without accounting for Ranuf and Saed being fixed.
• Including Saed in the coach choices when he is already assigned to the car.
• Excluding Ranuf from the coach when he is already assigned to the coach.
• Using permutations and multiplying by an order factor; this question asks for a set, not an ordered arrangement.
• Giving only the final answer without showing the combination; the mark scheme requires method marks.

Things to Be Careful About

• The coach already contains Ranuf, so you choose 34 more, not 35.
• Saed is not available for the coach, so the pool is 38, not 39 or 40.
• $^{38}C_{34}$ and $^{38}C_{4}$ are equal; either form is acceptable.
• The final numerical value is 73815.