There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is _______. [2024]

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Solution

Q.
There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is _______. [2024]

Ans.
(5626)

Group A Group B Ways

$4 m$ $4 w$ ${}^{4}{C_{4}} \cdot^{4} C_{4} = 1$

$3 m + 1 w$ $1 m + 3 w$ ${}^{4}{C_{3}} \cdot^{5} C_{1} \cdot^{5} {C_{1}}^{4} C_{3} = 400$

$2 m + 2 w$ $2 m + 2 w$ ${}^{4}{C_{2}} \cdot^{5} {C_{2}}^{5} {C_{2}}^{4} C_{2} = 3600$

$1 m + 3 w$ $3 m + w$ ${}^{4}{C_{1}}^{5} {C_{3}}^{5} {C_{3}}^{4} C_{1} = 1600$

$4 w$ $4 m$ ${}^{5}{C_{4}}^{5} C_{4} = 25$

Total number of ways = 1 + 400 + 3600 + 1600 + 25 = 5626

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