How many ways are there to place $5$ balls into $8$ boxes?

Question

$8$ boxes are arranged in a row. In how many ways can $5$ distinct balls be put into the boxes if each box can hold at most one ball and no two boxes without balls are adjacent?

Answer 1

HINTS:

Start by satisfying the claim that "no two empty boxes are adjacent".

For this you will need $4$ balls and I let you figure out in how many ways these can be arranged so that there are no empty boxes next to each other.

Finally you have $1$ ball left. How many empty boxes are left?

Hope this helped

If you need more guidance, please let me know.

Answer 2

As an alternative approach, first order the five balls left to right -- how many ways? Next, among the six spaces between adjacent balls or outside the balls on either side, pick three for the locations of the empty boxes -- how many ways? Multiply those two numbers.