JEE Mathematics 2023 Question with Solution

Given: Let the number of couples be $n$. The number of matches such that no couple is in a match is $840$.

Find: The total number of persons who participated in the tournament.

The total number of ways is given by

$$\frac{n(n-1)}{2} \cdot \frac{(n-2)(n-3)}{2} \cdot 2 = 840$$

Step 1: Simplify the equation:

$$n(n-1)(n-2)(n-3) = 840 \cdot 4$$ $$n(n-1)(n-2)(n-3) = 3360$$

Step 2: Factorize $3360$:

$$3360 = 8 \cdot 7 \cdot 6 \cdot 5$$

Thus, $n = 8$.

Step 3: Calculate the total number of persons:

$$\text{Total persons} = 2n = 2(8) = 16$$

Therefore, the number of persons is $16$.

Given: Let the total number of couples be $n$.

Find: The total number of players.

According to the given condition,

$${}^nC_2 \cdot {}^{n-2}C_2 \cdot 2 = 840$$

From this, we get $n = 8$. Hence the total number of players is

$$8 \times 2 = 16$$

Therefore, the total number of persons is $16$.