Question

A committee of $10$ is to be formed from $8$ women and $6$ men. In how many of these committees the women are in majority?

• A. $515$
• B. $545$
• C. $575$
• D. $595$

Answer 1

The correct option is D $595$

for majority, women should be $>5$ in no.

$\therefore$ selecting the committee $\Rightarrow$

${}^8{C}_6{\times }^6{C}_4{+}^8{C}_7{\times }^6{C}_3{+}^8{C}_8{\times }^6{C}_2$

$=\frac{8\times 7\times 6\times 5}{2\times 1\times 2\times 1}+\frac{8}{1}\times \frac{6\times 5\times 4}{3\times 2\times 1}+\frac{8}{8}\times \frac{6\times 5}{2\times 1}$

$=420+160+15$

$=595$