Question

In how many ways a committee of $5$ members to be formed out of $6$ boys and $4$ girls, if:
at least $2$ girls in the committee

Answer 1

$2$ girls

P($2$ girls)$=C_2^4$

P($3$ boys)$=C_3^6$

$C_2^4\times C_3^6$

$3$ girls

P($3$ girls)$=C_3^4$

P($2$ boys)$=C_2^6$

$C_3^4\times C_2^6$

$4$ girls

P($4$ girls)$=C_4^4$

P($1$ boy)$=C_1^6$

$C_4^4\times C_1^6$

No. of ways $=C_2^4\times C_3^6+C_3^4\times C_2^6$

$==\frac{4\times 3}{2}\times \frac{6\times 5\times 4}{6}+\frac{4}{1}\times \frac{6\times 5}{2}+1\times 6=6\times 20+60+6=120+66=186$