There are `m` men `and 2 women. Everyone plays two matches will all other participants. If number of matches played between men exceeds the number of matches played between men and women by 84, them `m` is equal to (A) `6` (B) `3` (C) `5` (D) `12`

Options

A. 12
B. 11
C. 9
D. 7

Answer

A

Explanation

Since, there are m - men and 2 - women and each participant plays two games with every other participant.
`therefore` Number of games played by the men between themselves `= 2 xx ""^(m)C_(2)`
and the number of games played between the men and the women `= 2 xx ""^(m)C_(1) xx ""^(2)C_(1)`
Now, according to the question,
`2 ""^(m)C_(2) = 2 ""^(m)C_(1) ""^(2)C_(1) + 84`
`rArr" " (m!)/(2!(m - 2)!) = m xx 2 + 42`
`rArr" " m(m - 1) = 4m + 84`
`rArr" " m^(2) - m = 4m + 84`
`rArr" " m^(2) - 5m - 84 = 0`
`rArr" " m^(2) - 12m + 7m - 84 = 0`
`rArr" " m(m - 12) + 7(m - 12) = 0`
`rArr" " m = 12" "[because m gt o]`