Question

Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from these 4 boxes so that from each box at least one red ball and one blue ball are chosen?

• A. 21816
• B. 85536
• C. 12096
• D. 156816

Answer 1

Answer: (A)

21816

Answer 2

Case-I : when exactly one box provides four balls ( 3 R 1 B or 2 R 2 B )
Number of ways in this case $5C_4 \times (3C_1 \times 2C_1)^3 \times 4$
Case-II : when exactly two boxes provide three balls ( 2 R 1 B or 1 R 2 B ) each
Number of ways in this case $(5C_3 - 1)^2 \times (3C_1 \times 2C_1)^2 \times 6$
Required number of ways =21816
Language ambiguity : If we consider at least one red ball and exactly one blue ball, then required number of ways is 9504 . None of the option is correct.
The correct option is (A): 21816