Mathematics 9709/53 — October/November 2021

Walkthrough

There are 26 members, but 3 of them (Mr Khan, Mrs Khan and Abad) must attend. Since the party has 20 places, once those 3 are guaranteed a place there are $20 - 3 = 17$ places left. These must be filled from the remaining $26 - 3 = 23$ members.

The selection is a combination because the order in which the 17 members are chosen does not matter. So we count the number of ways to choose 17 people from 23:

This is the same as choosing the 6 people who will not attend from the 23 non-family members, so it can also be written as $^{23}C_6$. Evaluating gives $100947$.

Key Takeaways

This question tests the idea of fixing required items before counting selections. When certain objects must be included, subtract them from both the total pool and the number being chosen, then use a combination for the remaining choices.

Common Mistakes

• Forgetting to subtract the 3 required members from the total, giving $^{26}C_{20}$.
• Choosing 20 people from the remaining 23 instead of 17, giving $^{23}C_{20}$.
• Using permutations ($^{23}P_{17}$) instead of combinations, because the order of selection is irrelevant.
• Not simplifying $^{23}C_{17}$ to $^{23}C_6$; this is not wrong but the final numerical answer must be correct.

Things to Be Careful About

• The three family members are distinct and already included, so they are not part of the choice.
• The problem can be viewed as choosing 6 people to exclude from the 23 non-family members, giving the same answer.
• Make sure the final answer is the integer $100947$, not just the expression $^{23}C_{17}$.
• The mark scheme awards M1 for the correct combination expression and A1 for the exact value.