# How many ways are there to place $5$ balls into $8$ boxes? [closed]

$$8$$ 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?

## closed as off-topic by B. Goddard, NCh, YiFan, Leucippus, Brian BorchersFeb 24 at 2:53

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## 2 Answers

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.

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.