Question

A committee of 3 members Is to be formed from 4 chartered accountants and 5 doctors associated with a social organization. In how many ways can the committee be formed such that,

(i) the chartered accountants are in majority?
(ii) the doctors are in majority?

Answer 1

There are 4 CA and 5 doctors. A committee of 3 members Is to be formed.

(i) Chartered accountants are in majority: In a committee of 3 members If chartered accountants are In majority then It can happen In the following two ways:
(a ) 2 CA and 1 doctor OR
(b ) 3 CA and no doctor
2 CA out of 4 and doctor out of 5 can be selected In ⁴C₂ × ⁵C₁ ways. OR
3 CA out of 4 and no doctor out of 5 can be selected in ⁴C₃ × ⁵C₀ ways.

Total combinations = ⁴C₂ × ⁵C₁ + ⁴C₃ × ⁵C₀
= \(\frac{4×3}{2×1}\) × 5 + 4 × 1
= 30 + 4 = 34

(ii) The doctors are in majority: In a committee of 3 members of the doctors are in majority then it can happen in the following two ways :
(a) 2 doctors and 1 CA OR
(b) 3 doctors and no CA
2 doctors out of 5 and one CA out of 4 can be selected in ⁵C₂ × ⁴C₁ ways OR
3 doctors out of 5 and no CA out of 4 can be selected in ⁵C₃ × ⁴C₀ ways.
Total combinations = ⁵C₂ × ⁴C₁ + ⁵C₃ × ⁴C₀
= \(\frac{5×4}{2×1}\) × 4 + \(\frac{5×4×3}{3×2×1}\) × 1
= 40 + 10 = 50