Question

A box contains \[5\] pairs of shoes. If \[4\] shoes are selected, then the number of ways in which exactly one pair of shoes obtained is:
A. \[120\]
B. \[140\]
C. \[160\]
D. \[180\]

Answer 1

Hint:
To solve this question, we will apply the permutation formula. For that total number of objects will be the total number of pairs of shoes, and for the number of choosing objects from the set, it will be the number of pairs of shoes selected, then after putting the given values in the formula, we will get our required answer.

Complete step by step solution:
We have been given that a box contains \[5\] pairs of shoes. It is given that \[4\] shoes are selected, we need to find the number of ways in which exactly one pair of shoes is obtained.

So, the number of pair of shoes the box contains \[ = {\text{ }}5\]
And the number of pair of shoes selected \[ = {\text{ }}4\]

Now to find exactly one pair of shoes, we will
apply the permutation, for that we will use the formula mentioned below

\[^n{P_r} = \dfrac{{n!}}{{(n - r)!}}\]
where, P = number of permutations
n = total number of objects
r = number of choosing objects from the set.

On putting the values in the above formula, we get
Number of ways of getting exactly one pair of shoes \[{ = ^5}{P_4}\]
$
   = \dfrac{{5!}}{{(5 - 4)!}} \\
   = \dfrac{{5!}}{{1!}} \\
   = 5 \times 4 \times 3 \times 2 \times 1 \\
   = 120 \\
$
So, the number of ways in which exactly one pair of shoes obtained is \[120.\]

Thus, option (A) \[120,\] is correct.

Note: In the solution, we have used the formula, \[^n{P_r} = \dfrac{{n!}}{{(n - r)!}}\], where P is the number of permutations, n is the total number of objects and r is the number of choosing objects from the set.
Students sometimes get confused when to apply permutation and when to apply combination, in order to find that first step is to ask yourself, whether in the question, the order of things is important or not, if it is important then it is a permutation question, if the order doesn't matter then it is a combination question.