Question

A boy has 8 trousers and 10 shirts. In how many ways can he select a shirt and a trouser.
(A) 80
(B) $8!\times 10!$
(C) 64
(D) $8{{!}^{2}}$

Answer 1

Hint: Here we solve this question by first finding the number of ways of selecting 1 shirt and then we find the number of ways of selecting 1 trouser using the formula, number of ways of selecting r objects from a set of n objects is ${}^{n}{{C}_{r}}=\dfrac{n!}{r!\times \left( n-r \right)!}$. Then we multiply the obtained values to find the total number of ways of selecting 1 shirt and 1 trouser from 8 trouser and 10 shirts.

Complete step-by-step solution:
We are given that the boy has 8 trousers and 10 shirts. We need to select 1 shirt and 1 trousers from them.
We need to find the number of ways of selecting 1 shirt and trouser.
First let us consider the shirts.
We need to find the number of ways of selecting 1 shirt from 10 shirts.
Now, let us consider the formula for selecting r objects from a set of n objects.
${}^{n}{{C}_{r}}=\dfrac{n!}{r!\times \left( n-r \right)!}$
Applying this formula for number of ways of selecting 1 shirt from 10 shirts is
$\begin{align}
  & \Rightarrow {}^{10}{{C}_{1}}=\dfrac{10!}{1!\times \left( 10-1 \right)!} \\
 & \Rightarrow {}^{10}{{C}_{1}}=\dfrac{10!}{9!}=10 \\
\end{align}$
So, 1 shirt can be selected from 10 shirts in 10 ways.
Now let us consider the trousers.
We need to find the number of ways of selecting 1 trouser from 8 trousers.
Applying the above formula for number of ways of selecting r objects from n objects, we get the number of ways of selecting 1 trouser from 8 trousers as,
$\begin{align}
  & \Rightarrow {}^{8}{{C}_{1}}=\dfrac{8!}{1!\times \left( 8-1 \right)!} \\
 & \Rightarrow {}^{8}{{C}_{1}}=\dfrac{8!}{7!}=8 \\
\end{align}$
So, we get the number of ways of selecting 1 trouser from 8 trousers is 8.
Now, we need to find the number of ways of selecting 1 shirt and 1 trouser from 10 shirts and 8 trousers, that is equal to
(Number of ways of selecting 1 shirt) $\times $ (Number of ways of selecting 1 trouser)
Substituting the values obtained above we get,
$\begin{align}
  & \Rightarrow 10\times 8 \\
 & \Rightarrow 80 \\
\end{align}$
So, the number of ways that the boy can select a shirt and a trouser is 80 ways. Hence, the answer is Option A.

Note: The common mistake that one makes while solving this problem is one might add the values obtained for number of ways of selecting 1 shirt and number of ways of selecting 1 trouser to find the total number of ways, that is one might solve it as,
Total number of ways = (Number of ways of selecting 1 shirt) + (Number of ways of selecting 1 trouser)
Then we get the answer as 10+8=18. But it is wrong. Here selecting the shirts and selecting the trousers does not affect each other. So, we need to multiply them, not add them.