The number of teams for a mixed doubles tennis match that can be formed from eight couples is
A. 784
B. 1820
C. 910
D. 1568

Hint: First we will use the that each team consists of are man and are woman, then we will find the total ways of selecting are both man and woman from 8 couples and then we will use the formula of combinations, \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\], where \[n\] is the total number of object and \[r\] is the number required in the above expression. Then we will find the total number of teams for match when one couple plays the match by using \[\dfrac{{{}^8{C_1} \times {}^8{C_1} \times {}^7{C_1} \times {}^7{C_1}}}{{2!}}\] and then simplify it.

Complete step by step answer:

We are given that there is a mixed double match from eight couples.

We know that each team consists of man and woman, we will find the total ways of selecting both are man and woman from 8 couples, we have
\[ \Rightarrow {}^8{C_1} \times {}^8{C_1}\]
Using the formula of combinations, \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\], where \[n\] is the total number of object and \[r\] is the number required in the above expression, we get

\[
   \Rightarrow \dfrac{{8!}}{{1!\left( {8 - 1} \right)!}} \times \dfrac{{8!}}{{1!\left( {8 - 1} \right)!}} \\
   \Rightarrow \dfrac{{8 \times 7!}}{{7!}} \times \dfrac{{8 \times 7!}}{{7!}} \\
   \Rightarrow 8 \times 8 \\
   \Rightarrow 64 \\
 \]

Finding the total number of teams for the match when one couple plays the match, we get
\[ \Rightarrow \dfrac{{{}^8{C_1} \times {}^8{C_1} \times {}^7{C_1} \times {}^7{C_1}}}{{2!}}\]

Using the formula of combinations again in the above equation to find the number of teams, we get
\[
   \Rightarrow \dfrac{{\dfrac{{8!}}{{1!\left( {8 - 1} \right)!}} \times \dfrac{{8!}}{{1!\left( {8 - 1} \right)!}} \times \dfrac{{7!}}{{1!\left( {7 - 1} \right)!}} \times \dfrac{{7!}}{{1!\left( {7 - 1} \right)!}}}}{{2!}} \\
   \Rightarrow \dfrac{{\dfrac{{8 \times 7!}}{{7!}} \times \dfrac{{8 \times 7!}}{{7!}} \times \dfrac{{7 \times 6!}}{{6!}} \times \dfrac{{7 \times 6!}}{{6!}}}}{{2!}} \\
   \Rightarrow \dfrac{{8 \times 8 \times 7 \times 7}}{2} \\
   \Rightarrow \dfrac{{64 \times 49}}{2} \\
   \Rightarrow 1568 \\
 \]

Hence, option D is correct.

Note: In solving this question, students should note here that while reading this problem one has to take care of all the steps to find the final answer. One should know the right time to use the formula of permutations or combinations, as a combination is a selection of an item from a collection, such that the order of selection does not matter, but a permutation is an arrangement of its member into a sequence or linear order. Students must avoid calculation mistakes. We should substitute the values in the formula properly.