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  4. A committee of 4 persons is to be formed from 2 ladies, 2 old men and 4 young men such that it includes at least 1 lady, at least 1 old man and at most 2 young men. Then the total number of ways in which this committee can be formed is:

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Q. A committee of $4$ persons is to be formed from $2$ ladies, $2$ old men and $4$ young men such that it includes at least $1$ lady, at least $1$ old man and at most $2$ young men. Then the total number of ways in which this committee can be formed is:

JEE MainJEE Main 2013Permutations and Combinations

Solution:

Required number of ways
$=^{ 2}C_{1} \times ^{2}C_{1} \times ^{4}C_{2} + ^{2}C_{1} \times ^{2}C_{2} \times ^{4}C_{1} + ^{2}C_{2}\times ^{ 2}C_{1} \times ^{4}C_{1} + ^{2}C_{2} \times ^{2}C_{2} \times ^{4}C_{0}$
$= 2\times2\times \frac{4\times 3}{2}$
$+2\times 1\times 4+1\times 2\times 4+1\times 1\times 1$
$= 24 + 8 + 8+1=41$

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