Verified Solution • 2025–26 Session moderate

QJEEM23Ma0781: JEE Main 2023 Mathematics - Class 11 JEEM23Ma0781 Solution: Some couples participated in a mixed doubles badminton…

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 ___________.

Step-by-Step Verified Solution

Let, nn be the total number of couples who participated in the tournament.

According to the question, 2×nC2×n−2C2=8402 \times{ }^n C_2 \times{ }^{n-2} C_2=840

amp;⇒nC2×n−2C2=420amp;⇒n!2!(n−2)!×(n−2)!(n−4)!2!=420amp;⇒n(n−1)(n−2)(n−3)4=420\begin{aligned}& \Rightarrow{ }^n C_2 \times{ }^{n-2} C_2=420 \\\\& \Rightarrow \frac{n !}{2 !(n-2) !} \times \frac{(n-2) !}{(n-4) ! 2 !}=420 \\\\& \Rightarrow \frac{n(n-1)(n-2)(n-3)}{4}=420\end{aligned}

Put n=8n=8 satisfied the equation.

So, n=8n=8

Hence, total number of players who participated
in the tournament =2×8=16=2 \times 8=16

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