A committee of 10 persons is to be formed from a group of 10 women and 8 men. How many possible committees will have at least 5 women?

Question

A committee of 10 persons is to be formed from a group of 10 women and 8 men.

(i) How many possible committees will have at least 5 women?

(ii) How many possible committees will have men in majority?

Answer 1

(i) A committee of 10 persons is to be formed from 10 women and 8 men such that the committee contains at least 5 women.

Consider the following table:

∴ Number of committees with at least 5 women

= 14112 + 14700 + 6720 + 1260 + 81

= 36873

(ii) Number of committees with men in majority

= Total number of committees – (Number of committees with women in majority + women and men equal in number)

= ¹⁸C₁₀ - 36873

 = ¹⁸C₈ - 36873

 = 43758 - 36873

 = 6885