Question

There are 12 seats in a row. How many ways can we seat 4 students if no two students are to sit in adjacent seats?

Answer 1

We have 12 seats and 4 students.

So, if the students are sitted and we look from left to right, so the seat to the right of every student except the last one needs to be empty. So, the answer is the same as if the 4 students need to occupy 12-(4-1)=12-3=9 seats without any restriction.

So, basically we need to find the number of ways in which 4 students can sit in 9 seats.

That is, we need to find .

We know,

So, 4 students (if they cannot be distinguished from each other) can be seated in 12 seats so that no two students sit in adjacent seats in ways.

Now, if the students can be distinguished from each other, then the final answer will be multiplied with the number of ways of arranging the different students, that is 4!.

So, we will get

So, 4 students (if they can distinguished from each other) can be seated in 12 seats so that no two students sit in adjacent seats in ways.