How many ways can 7 identical balls be distributed in 5 different boxes if any box can contain any number of balls?

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

Attempt: Choose the pair of boxes without balls first. There are $7$ ways to do this.

Then choose the other box without balls. There are $6$ ways to do this.

Then subtract the number of ways which the $3$ boxes are together. There are $6$ ways to do this.

Thus there are $36$ ways to choose empty boxes, and $5!$ ways to choose how to put in the balls, so we get $4320$.

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There are 5 different boxes and 7 different balls. All the 7 balls are to be distributed among the 5 boxes placed in a row so that any box can receive any number of balls.In how many ways can these balls be distributed so that box 2 and box 4 contain only 1 and 2 balls respectively?

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