Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are

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Gujarati

Hindi

6.Permutation and Combination

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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?

$21816$

$85536$

$12096$

$156816$

(IIT-2022)

Solution

$Case-I$ : 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.

Standard 11

Mathematics

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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?

Solution

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