Question

There are 10 pairs of shoes in a cupboard from which 4 shoes are picked at random. If p is the probability that there is at least one pair, find 323p.

Answer 1

Hint: Here, first of all we will find the total number of ways of choosing 4 shoes from a total of 10 pairs of shoes. Then, we will find the number of ways of not selecting a pair. Thereafter, we will find the probability of not getting a pair using these two values. At last we will subtract the obtained value from 1 to find the probability of getting at least one pair of shoes.

Complete step-by-step answer:
Here, we are given 10 pairs of shoes. Since, each pair contains 2 shoes, so 10 pairs will contain 20 shoes.
Now, let us find the number of ways of choosing 4 shoes from a total of 20 shoes.
For the first time, we can select a shoe in 20 ways.
The second time, the number of shoes gets reduced by 1, so we can select a shoe in 19 ways.
Similarly, we have 18 ways in the third time and 17 ways in the fourth time.
Therefore, by the fundamental principle of counting, the total number of ways of choosing 4 shoes out of 10 pairs or 20 shoes = $20\times 19\times 18\times 17$.
Now, we will find the number of ways of selecting 4 shoes such that no pair is selected.
In the first time, the total number of ways of selecting a shoe = 20.
Now, from the remaining 19 shoes, there will be a shoe which is the pair of the shoe already selected. That is why we are left with 18 choices.
Similarly, for choosing the third and fourth shoe, we have 16 and 14 choices respectively.
Therefore, the number of ways in which no pair is selected = $20\times 18\times 16\times 14$.
Thus, the probability of not getting a pair is given as:
$\dfrac{20\times 18\times 16\times 14}{20\times 19\times 18\times 16}=\dfrac{224}{323}$
Hence, the probability of getting at least one pair is:
$\begin{aligned} & =1-\dfrac{224}{323} \\\ & =\dfrac{323-224}{323} \\\ & =\dfrac{99}{323} \\\ \end{aligned}$
So, the value of 323p = $323\times \dfrac{99}{323}=99$.

Note: Students should note here that we are given that p is the probability of getting at least one pair which means that it is possible that we get more than one pair. So, we find the probability of not getting a pair and subtracted it from 1. A mistake that a student may make is taking a number of choices incorrectly while finding the number of ways of not getting a pair. The previously selected shoe will have its pair in the remaining shoes and hence it should not be considered.