Question

A box contains two white, three black and four red balls. The number of ways in which we can select three balls from the box if at least one black ball is to be included in the selection is –

• A. 64
• B. 104
• C. 60
• D. 62

Answer 1

Answer: (A)

64

Answer 2

If there is no restriction, we can select 3 balls out of (2 + 3 + 4) balls in ⁹C₃ ways. If black balls are to excluded, this can be done in ⁶C₃ ways. Therefore the number of ways when at least one black ball is included is

⁹C₃ – ⁶C₃ = $\frac{9 \times 8 \times 7}{3 \times 2 \times 1}$– $\frac{6 \times 5 \times 4}{3 \times 2 \times 1}$= 84 – 20 = 64.