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  4. 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?

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Q. 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?

JEE AdvancedJEE Advanced 2022

Solution:

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Case-I : when exactly one box provides four balls ( $3 R 1 B$ or $2 R 2 B$ )
Number of ways in this case ${ }^5 C _4\left({ }^3 C _1 \times{ }^2 C _1\right)^3 \times 4$
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 $\left({ }^5 C _3-1\right)^2\left({ }^3 C _1 \times{ }^2 C _1\right)^2 \times 6$
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.