Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is 840, then the total number of persons, who participated in the tournament, is ___________.

Solution

Maths sir · 4 steps

Pick the two men first, then pick the two women only from the other couples.

1. Read the condition. Say there are n couples, so n men and n women. A mixed doubles match has 2 men and 2 women. No couple in a match means no man is on court with his own wife, whether as partner or as opponent.
2. Count one match. Choose 2 men in C(n, 2) ways. Their wives are now ruled out, so choose 2 women from the other n-2 in C(n-2, 2) ways. The 2 men and 2 women can form teams in 2 ways: M₁ with W₁ or M₁ with W₂.

   2C(n, 2)C(n-2, 2) = n(n-1)(n-2)(n-3)/2 = 840

3. Solve for n. So n(n-1)(n-2)(n-3) = 1680 = 8 × 7 × 6 × 5. That is a product of four consecutive integers, so n = 8. Check: 2 × 28 × 15 = 840.
4. So. There are 8 couples, and the number of persons is 2 × 8 = 16.

The common slip: 8. Stopping at n = 8 gives the number of couples. Each couple is two people, so the answer is 16.

The idea to keep. When some pairs are forbidden together, choose one group first and then choose the rest from what is still allowed. Do not forget to count how the chosen players can be split into teams.

Answer from the NTA final answer key. Solution by PrepWiser.