Question

A box contains 5 red and 4 blue balls. In how many ways can 4 balls be chosen such that there are at most 3 balls of each colour?

• A. 132
• B. 242
• C. 60
• D. 120
• E. 240

Hint:

Break “at most” problems into valid cases and exclude the disallowed extremes.

Answer 1

The Correct Option is D

Step 1: Restriction.
“At most 3 of each colour” → exclude cases of 4 red and 4 blue.
Step 2: Valid cases.
- 3 red + 1 blue. - 2 red + 2 blue. - 1 red + 3 blue.
Step 3: Count each case.
- 3R,1B: \(\binom{5}{3} \binom{4}{1} = 10 \times 4 = 40\).
- 2R,2B: \(\binom{5}{2} \binom{4}{2} = 10 \times 6 = 60\).
- 1R,3B: \(\binom{5}{1} \binom{4}{3} = 5 \times 4 = 20\).

Step 4: Add.
Total = \(40 + 60 + 20 = 120\).
Final Answer:
\[ \boxed{120} \]