# Find the no. of ways in which 6 boys and 6 girls can be seated in a row such that no two girls are seated together

My approach to this problem was as follows - Seat all the boys in a row before the girls. Then arrange the 6 girls in the 7 places between the boys(5 places between the boys and 2 places at the extremes). Therefore, the answer will come out to be $$6! * 7!$$ .

What is written in the solutions - Seat all the 6 girls before the boys in a row. Now find the number of ways of seating 6 boys in the 5 places between the girls. For the one boy left, the number of places to sit in is 12. Therefore, the answer comes out to be $$6! * 6! * 12$$. What is the cause of these two conflicting answers? Is my approach wrong or is the approach given in the book wrong?

• @TahaDirek But in my approach, there is one place left as there are 7 places and only 6 girls. That one place can fall in between 2 boys and hence my answer includes the cases where 2 boys are sitting together. – Pratham Yadav Mar 16 '20 at 12:10
• Sorry, I've misread. – Taha Direk Mar 16 '20 at 12:20
• The answer of $6!\cdot 7!$ is correct. The answer of $6!\cdot 6!\cdot 12$ is incorrect. To see why, look at what happens for if two boys are between two girls. Using lowercase for girls and uppercase for boys, you might have had the scenario $aABb\cdots$ where $A$ was placed between $a$ and $b$ in the first step and then $B$ was placed after $A$, or you might have had the situation $aABb\cdots$ where $B$ was placed between $a$ and $b$ and then $A$ was placed before $B$ in the second step. – JMoravitz Mar 16 '20 at 12:31
• @JMoravitz Thanks. I've got it now. Such types of cases are difficult to visualize and it becomes difficult to find out which approach is the correct one. – Pratham Yadav Mar 16 '20 at 12:35

## 1 Answer

First find which positions on the bench can be occupied by the girls. One solution is positions 1,3,5,7,9,11. Or there could be, starting from this seating, one more boy before the first,2nd, 3rd ,4th 5th or 6th girl. So there are exactly 7 possibilities for which positions can be occupied by girls. For each of these seating possibilities the girls can be placed in 6! ways and the boys in 6! ways. So the answer is 7x6!x6!=7!x6!.