Question

A committee of 5 is to be formed out of 6 men and 4 ladies. In how many ways can this be done when
(i) at least 2 ladies are included.
(ii) at most 2 ladies are included.

Answer 1

$5⟶$ To be choosen

$6$ mens $4$ ladies

(1) Atleast $2$ ladies

There can be $2$ ladies , $3$ or even $4$

${\Rightarrow }^4{P}_2{\ast }^6{P}_3{+}^4{P}_3{\ast }^6{P}_2{+}^4{P}_4{\ast }^6{P}_1\Rightarrow 12\times 120+120+144\Rightarrow 1440+264\Rightarrow 1704$

So $1704$ ways are possible.

(2) Almost $2$ Ladies

There can be one or two

${\Rightarrow }^4{P}_1{\ast }^6{P}_4{+}^4{P}_2{\ast }^6{P}_3\Rightarrow 1440+144\Rightarrow 1584$

Hence $1584$ ways are possible.