Question

6 boys and 6 girls sit in a row randomly. Find the probability that all the girls sit together.
A.$\dfrac{1}{64}$
B.$\dfrac{1}{8}$
C.$\dfrac{1}{132}$
D.None of these

Answer 1

Hint: In this question we are given 6 boys and 6 girls sitting in a row randomly. First, we evaluate the total number of permutations possible for various sitting patterns. Then, we evaluate permutations possible for all girls to sit together. To obtain the probability we divide both the values.

Complete step-by-step answer:
In mathematics, a permutation is defined as the arrangement into a sequence or linear order, or if the set is already ordered, a rearrangement of its element.
As given in the question, there are 6 boys and 6 girls sitting in a row randomly.
Then, the total number of people = 6 boys + 6 girls =12.
So, the arrangement of 12 people$=12!$.
All the girls sit together, so we consider all 6 girls to be one 1 group of girls.
Now, we have 6 boys and 1 group of girls = 7 people.
Number of arrangements $=7!$.
Now, girls can arrange themselves in the group =$6!$
So, total permutations for this arrangement $=7!\times 6!$
The probability of an event is defined as, $P=\dfrac{\text{favourable outcomes}}{\text{total outcomes}}$.
The probability when all girls sit together is $=\dfrac{7!\times 6!}{12!}$
The probability of the event $=\dfrac{1}{132}$
Therefore, the probability that all the girls sit together is $\dfrac{1}{132}$.
Therefore, option (c) is correct.
Note: Students must be careful while calculating the permutations when restrictions are imposed. They must remember that girls can also permute in 6! ways. So, total permutation must be the multiplication of both the permutations.