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Q.
Number of ways in which a lawn-tennis mixed double be made from seven married couples if no husband and wife play in the same set is
Permutations and Combinations
Solution:
We first select $2$ men out of $7$ in $^7C_2$ ways. Now we exclude the wives of these two selected men and so select $2$ ladies from remaining $5$ ladies in $^5C_2$ ways. Let $A, B$ be two men and $X, Y$ be the ladies playing in one set. Then we can have
(i) $A$ and $X$ plying against $B$ and $Y$.
(ii) $A$ and $Y$ playing against $B$ and $X$.
Then the total number of ways is $^7C_2 \times\,{}^5C_2 \times 2 = 21 \times 10 \times 2
= 420$.