# Six boys and six girls are to sit in a row at random. The probability that all the 6 girls sit together is

A. $\dfrac{{6!6!}}{{12!}}$

B. $\dfrac{{6!7!}}{{12!}}$

C. $\dfrac{{2 \times 6!6!}}{{12!}}$

D. $\dfrac{{2 \times 6!7!}}{{12!}}$

Last updated date: 24th Mar 2023

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Answer

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Hint: Here we will use the formula of permutation to find the total numbers of events and the favourable events and then we will find the probability.

Complete step-by-step answer:

First, we will find the total number of events. Given that there are six boys and six girls so that there are a total 12 persons who are said to sit in a row. Now, using the formula ${}^n{P_r}$, we will find the total number of ways 12 persons can sit.

So, total number of ways of sitting = ${}^n{P_r}$ =\[\dfrac{{n!}}{{(n - r)!}}\], where n is the number of persons and r is the total places. Here in this question, n = 12 and r = 12. So, applying the above formula.

Total number of ways = ${}^{12}{P_{12}} = \dfrac{{12!}}{{(12 - 12)!}} = \dfrac{{12!}}{{0!}} = 12!$

Now, let a group be made in which all girls sit together. Let take this group as one element. So, we have a total 7 elements left out of which there are six boys and a group of girls. So, we have to find a number of ways in which we can arrange these 7 elements in a row.

So, number of ways = ${}^7{P_7} = 7!$

Now the girls in the group can be seated in any arrangement. So, number of ways in which girls can be seated in the group = ${}^6{P_6} = 6!$

So, the total number of ways in which all girls sit together = 6! x 7!

So, using the probability formula,

Probability = favourable events / Total number of events

Here the favourable events are the number of ways in which all girls sit together. Total number of events is the number of ways in which all 12 persons sit in a row.

So, required probability = $\dfrac{{6!7!}}{{12!}}$

So, the answer is option (B).

Note: In such types of problems, one big mistake which is done most of the time is that students forgot to take all the cases. They don’t apply the formula correctly and make a mistake which eventually leads to the wrong answer. So, it is necessary that you take all the cases and carefully apply the formula of permutation to find the total number of arrangements. Also, apply the proper formula of probability without a mistake.

Complete step-by-step answer:

First, we will find the total number of events. Given that there are six boys and six girls so that there are a total 12 persons who are said to sit in a row. Now, using the formula ${}^n{P_r}$, we will find the total number of ways 12 persons can sit.

So, total number of ways of sitting = ${}^n{P_r}$ =\[\dfrac{{n!}}{{(n - r)!}}\], where n is the number of persons and r is the total places. Here in this question, n = 12 and r = 12. So, applying the above formula.

Total number of ways = ${}^{12}{P_{12}} = \dfrac{{12!}}{{(12 - 12)!}} = \dfrac{{12!}}{{0!}} = 12!$

Now, let a group be made in which all girls sit together. Let take this group as one element. So, we have a total 7 elements left out of which there are six boys and a group of girls. So, we have to find a number of ways in which we can arrange these 7 elements in a row.

So, number of ways = ${}^7{P_7} = 7!$

Now the girls in the group can be seated in any arrangement. So, number of ways in which girls can be seated in the group = ${}^6{P_6} = 6!$

So, the total number of ways in which all girls sit together = 6! x 7!

So, using the probability formula,

Probability = favourable events / Total number of events

Here the favourable events are the number of ways in which all girls sit together. Total number of events is the number of ways in which all 12 persons sit in a row.

So, required probability = $\dfrac{{6!7!}}{{12!}}$

So, the answer is option (B).

Note: In such types of problems, one big mistake which is done most of the time is that students forgot to take all the cases. They don’t apply the formula correctly and make a mistake which eventually leads to the wrong answer. So, it is necessary that you take all the cases and carefully apply the formula of permutation to find the total number of arrangements. Also, apply the proper formula of probability without a mistake.

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