Question

A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw?

• A. 84
• B. 60
• C. 129
• D. 114

Hint:

When calculating the number of ways to select items with certain conditions, use the complementary counting technique by subtracting the cases that don't satisfy the condition.

Answer 1

The Correct Option is A

Step 1: Calculate total number of ways to choose 3 balls.
Total number of ways to choose 3 balls from 9 is \( \binom{9}{3} \).
Step 2: Subtract the cases with no black balls.
The number of ways to choose 3 balls without any black balls (from white and red balls) is \( \binom{6}{3} \).
Step 3: Conclusion.
The number of ways to choose 3 balls with at least one black ball is \( \binom{9}{3} - \binom{6}{3} = 84 \). Final Answer: \[ \boxed{84} \]