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$8$ 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?

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

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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.

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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.

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