Question

In how many ways can a committee of 5 persons be formed out of 6 men and 4 women when at least one woman has to be necessarily selected ?

Answer 1

Total men = 6

Total women = 4

Total persons in committee = 5

(where at least are women has to be selected)

This can be done in

${}^4{C}_1{\times }^6{C}_4{+}^4{C}_2{\times }^6{C}_3{+}^4{C}_3{\times }^6{C}_2{+}^4{C}_4{\times }^6{C}_1$

${(}^n{C}_r=\frac{n!}{r!(n-r)!}){(}^n{C}_r=1{,}^n{C}_1=n)$

$=\left(\frac{4\times 6!}{4!\times 2!}\right)+(\frac{4!}{2!2!}\times \frac{6!}{3!3!})+(\frac{4!}{3!1!}\times \frac{6!}{2!4!})+(1\times 6)$

$=\left(\frac{4\times 6\times 5}{2}\right)+(\frac{4\times 3}{2}\times \frac{6\times 5\times 4}{3\times 2})+\left(\frac{4\times 6\times 5}{2}\right)+\left(6\right)$

$=\left(60\right)+\left(120\right)+60+6$

= 246 ways