total number of ways At a cafeteria, there are $10$ seats in a row, and $10$ people are lined up to walk into the cafeteria. The first person can sit anywhere, but any future person will only sit in a seat next to someone who is already sitting down. If the first person sits in the $5$th seat, how many total ways are there for the rest of the people to sit down?

Since we have a restriction that the next person will only sit in a seat next to someone who is already sitting down and there's only one person seated initially then at any moment the persons already seated will be seated in a contiguous sequence of chairs.

Given this, we can think about the state at a given point to be defined by the chair closest to the left that have a person seated, L, and the chair closest to the right that also have a person seated, R, forming a pair (L,R).

We start with the state (5,5). From here we have two restrictions: L must be positive and R must be at most 10. So, if we see how the people seat sequentially and set a '0' if a person seats at the left and set a '1' if a person seats at the right, the resulting binary string must have size 9 and be composed with exactly 4 zeroes and 5 ones.

The number of such binary strings is: (9C4)=126.