I have this question: How many seating arrangements are there for $5$ people to sit in $10$ seats in a row, when $2$ people can't sit next to each other?
My idea:
If there must be at least one space between every 2 people, the spaces must be something like this:
_ _ s _ s _ s _ s _
There must be $2$ open seats next to each other, so they have $5$ options of where to be, and the person who sits there, have $2$ options to choose from.
The seating arrangements for $5$ people is $5!$ in a standard row,
so overall: $5 \cdot 2 \cdot 5! = 1200$.
My answer is wrong, so I was wondering what is a better way to think about it.
Best Answer
You must have
p_p_p_p_pand one more empty seat somewhere. There are $6$ possible locations for the remaining empty seat: on one end, or between two people. The $5$ people can be arranged in $5!$ different ways in their chosen seats, so altogether there are $6\cdot5!=720$ arrangements.Notice that you don’t have to have two empty seats next to each other: you can have, for instance,
_p_p_p_p_p. If you want to use that line of argument, you should notice that there are just $4$ places between two people, so there are $4$ places to put the pair of seats. But then there are also the arrangements_p_p_p_p_pandp_p_p_p_p_, for a total of $6$.