There is a bag that contains 4 yellow balls, 5 black balls and 7 pink balls. The number of ways in which three balls can be drawn from the box so that at least one of the balls is black is:

Question: There is a bag that contains 4 yellow balls, 5 black balls and 7 pink balls. The number of ways in which three balls can be drawn from the box so that at least one of the balls is black is:

Solution:
Total number of balls = 16
The required number of ways
Case 1:
1 black and 2 others = ⁵C₁ × ¹¹C₂
= 5 × 55
= 275

Case 2:
2 black and 1 other = ⁵C₂ × ¹¹C₁
= 10 × 11
= 110

Case 3:
All the three black = ⁵C₃ = 10

Total = 275 + 110 + 10 = 395 ways
Hence option number (1) is the right answer