Question

The number of ways of selecting \[15\] teams from \[15\] men and \[15\] women, such that each team consists of a man and a woman, is:
A). \[1960\]
B). \[1880\]
C). \[1120\]
D). \[1240\]

Answer 1

Hint: In the given question, we have been given that there are some given number of men and women. We have to make some given number of teams from these given numbers of players where each consists of a given number of men and women. We are going to solve it by adding the individual ways of choosing each pair using the formula of Combination (as in Permutation and Combination).
Formula used:
We have to calculate the value of a combination using the formula,
\[^n{C_m} = \dfrac{{n!}}{{\left( {n - m} \right)!m!}}\]

Complete step by step solution:
We have to make teams of one man and one woman.
This is done by using the formula of Combination and adding them together, so we have,
\[\sum\limits_{n = 15}^1 {^n{C_1}} {.^n}{C_1}\]
So, the total number of ways is:
\[{ = ^{15}}{C_1}{.^{15}}{C_1}{ + ^{14}}{C_1}{.^{14}}{C_1} + ...{ + ^1}{C_1}{.^1}{C_1}\]
We know, \[^n{C_1} = \dfrac{{n!}}{{\left( {n - 1} \right)!1!}} = n\]
Hence, we have,
\[ = 15 \times 15 + 14 \times 14 + ... + 1 \times 1 = {15^2} + {14^2} + ... + {1^2}\]
Now, we know, sum of square first \[n\] natural numbers is given by,
\[S = \dfrac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}\]
So, putting \[n = 15\], we have,
\[ = \dfrac{{15\left( {15 + 1} \right)\left( {2 \times 15 + 1} \right)}}{6} = \dfrac{{15 \times 16 \times 31}}{6} = 1240\]
Hence, the correct option is D.

Additional Information:
Here, we calculated the value of a combination. But, if we had a permutation, then we would have used the formula:
\[^n{P_m} = \dfrac{{n!}}{{\left( {n - m} \right)!}}\]

Note: In the given question, we had to calculate the number of ways of choosing a team of men and women from the given number of players. We solved it by adding all the possibilities of such arrangement and got our answer. We just need to remember all the formulae and know their basic meaning about when one is used and when the other is used.