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6.Permutation and Combination
hard
A group of students comprises of $5$ boys and $n$ girls. If the number of ways, in which a team of $3$ students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is $1750$, then $n$ is equal to
A
$24$
B
$28$
C
$27$
D
$25$
(JEE MAIN-2019)
Solution
Given $5$ boys and $n$ girls
Total ways of farming team of $3$ Members under given condition
${ = ^5}{C_1}{.^n}{C_2}{ + ^5}{C_2}{.^n}{C_1}$
${ \Rightarrow ^5}{C_1}{.^n}{C_2}{ + ^5}{C_2}{.^n}{C_1} = 1750$
$ \Rightarrow \frac{{5n(n – 1)}}{2} + 10n = 1750$
$ \Rightarrow \frac{{n(n – 1)}}{2} + 2n = 350$
$ \Rightarrow {n^2} + 3n = 700$
$ \Rightarrow {n^2} + 3n – 700 = 0$
$ \Rightarrow n = 25$
Standard 11
Mathematics
