A car can hold 3 people in the front seat and 4 in the back seat. In how many ways can 7 people be seated in the car if 2 particular people must sit in the back seat and 1 particular person is the driver?
A car can hold 3 people in the front seat and 4 in the back seat. In how many ways can 7 people be seated in the car if 2 particular people must sit in the back seat and 1 particular person is the driver?
The driver is fixed. That means there are six people left.
Since two of the six insist on sitting in the back, we have only four people from which can choose the other two for the front seat. $\displaystyle \binom {4} {2}$ and two ways to arrange those two. At this point we have four people remaining for the back seat, $\displaystyle 4!$.
Or could you fix the $\displaystyle 2 $ particular people in the back seat? That means there are $\displaystyle 5 $ people left. $\displaystyle 1 $ must be the driver. So we have $\displaystyle 4 $ people from which we can choose the other $\displaystyle 2 $ for the front seat. Same answer: $\displaystyle 2\cdot 4! \binom{4}{2} $.