A committee of 5 members is to be formed by selecting out of 4 men and 5 women. In how many different ways the committee can be formed if it should have at least 1 man?

Answer: Option C

The committee should have
(1 man, 4 women) or (2 men, 3 women) or (3 men, 2 women) or ( 4 men, 1 woman)
Required number of ways
$$=\left({}^4{C}_1\times {}^5{C}_4\right)+\left({}^4{C}_2\times {}^5{C}_3\right)$$      $$+\left({}^4{C}_3\times {}^5{C}_2\right)$$   $$+\left({}^4{C}_6\times {}^5{C}_1\right)$$
$$=\left({}^4{C}_1\times {}^5{C}_1\right)+\left({}^4{C}_2\times {}^5{C}_2\right)$$      $$+\left({}^4{C}_1\times {}^5{C}_2\right)$$   $$+\left({}^4{C}_4\times {}^5{C}_1\right)$$
$$=\left(4\times 5\right)+\left(\frac{4\times 3}{2\times 1}\times \frac{5\times 4}{2\times 1}\right)$$      $$+\left(4\times \frac{5\times 4}{2\times 1}\right)$$   $$+\left(1\times 5\right)$$
$$=\left(20+60+40+5\right)$$
$$=125$$