Question

A select a group of 4 members and is to be formed from 8 men and 6 women in such a way that the group must have at least 1 woman. In how many different ways can it be done?
1. 264
2. 728
3. 931
4. 1001

Answer 1

Correct Answer - Option 3 : 931

Formula used:

If we have a group of 'n' objects out of which we make a selection taking 'r' object at a time, then the number of such selections is given by the combination formula

nCr = n!/(r!(n – r!))

Concept used:

At least 1 women in group means, there can be 1, 2, 3 or 4 women in the group

Calculation:

The different combination of men and women to form the group

3 men and 1 woman + 2 men and 2 women + 3 women and 1 man + 4 women

The selection of required man and women from 8 men and 6 women

⇒ ⁸C₃ × ⁶C₁ + 8C₂ × 6C₂ + 8C₁ × 6C₃ + 6C₄

⇒ 8!/(3! × (8 – 3)!) × 6!/(1! × (6 – 1)!) + 8!/(2! × (8 – 2)!) × 6!/(2! × (6 – 2)!) + 8!/(1! × (8 – 1)!) × 6!/(3! × (6 – 3)!) + 6!/(4! × (6 – 4)!)

⇒ 8!/(3! × 5!) × 6!/(1! × 5!) + 8!/(2! × 6!) × 6!/(2! × 4!) + 8!/(1! × 7!) × 6!/(3! × 3!) + 6!/(4! × 2!)

⇒ (8 × 7 × 6 × 5!)/(3 × 2 × 5!) × (6 × 5!)/5! + (8 × 7 × 6!)/(2 × 6!) × (6 × 5 × 4!)/(2 × 4!) + (8 × 7!)/7! × (6 × 5 × 4 × 3!)/(3! × 3 × 2) + (6 × 5 × 4!)/(4! × 2)

⇒ (8 × 7) × 6 + (4 × 7) × (3 × 5) + 8 × (5 × 4) + (3 × 5)

⇒ 336 + 420 + 160 + 15 = 931

∴ The number of ways to arrange men and women is 931