Question

In how many ways can a mixed double tennis game be arranged from 7 married couples. If no husband and wife play in the same game ?
A. 28
B. 70
C. 210
D. 420

Answer 1

Hint: In this particular question when we select two men from 7 couples then we can’t select two wives from the 7 womens because they can be the wife of selected men. So, we had to select wives from the 5 women (excluding wives of 2 men). And this could be done in reverse order also. Like first we selected two womens from 7. And after that select two men from the 5 men. And these men and women should be selected using the formula \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\] .

Complete step-by-step answer:
Now let us assume the 1st two husbands as X and Y and excluding their wives we had to assume two other ladies let say A and B out of 5 remaining ladies.
So, we can represent the 2 husbands with the formula as \[{}^7{C_2}\] and the 2 assumed ladies ( A and B ) can be represented as \[{}^5{C_2}\] because we cannot count the wives of X and Y(here).
Now these \[{}^7{C_2}\] husbands can team up with other ladies in
\[{}^7{C_2} = \dfrac{{7!}}{{2!\left( {7 - 2} \right)!}} = 21ways\]
And these \[{}^5{C_2}\] ladies can team up with other gents apart from their husband in
\[{}^5{C_2} = \dfrac{{5!}}{{2!\left( {5 - 2} \right)!}} = 10ways\]
So the total number of ways of selecting the players for mixed match = \[{}^7{C_2}\] * \[{}^5{C_2}\]
\[ \Rightarrow 21 \times 10 = 210ways\]
Now since we have calculated these number for men first and women second , we could also do this in reverse order .Therefore we have to multiply the obtained value with 2.
So, now required number of ways is 2 * 210 = 420.
Hence D is the correct option.

Note: In such type of questions in which we have to pair up or we have to team up the players by the formula of permutation and combinations \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\] are very helpful and make the solution easy and quickly as compared to the solution in other ways. And as here it is mentioned that it is a doubles tennis which means there must be 2 members in a team but the team members cannot be a couple by keeping these main points of question in mind we can solve the question.