In how many ways can I put 1 to 5 mirrors in 8 rooms?

Question

A queen has 8 rooms and 12 indistinguishable mirrors, how many ways are there to hang these mirrors in 8 rooms such that every room has at least 1 mirror?

Answer 1

Let the numbers of mirror in a room be $x_i$, where $x_i$ is a positive integer.

We need to find solutions for,

$$x_1+x_2+x_3+x_4+x_5+x_6+x_7+x_8=12$$

This can be seen as 12 stars and 7 bars.

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Now there are 11 gaps between 12 stars and you have 7 bars to place, where each bar separates number of mirrors in a room.

So the answer will be?

Answer 2

We have 12 mirrors, but 8 of them must be allocated so each room has at least 1. So all we care about is placing the remaining 4 mirrors in 8 rooms.

This is a perfect place to use stars and bars, since we have 4 mirrors to spread across 8 rooms. This leads us to find the total number of combinations as $${8+4-1\choose4} ={11\choose4} =\color{red}{330}$$

If you are confused as why $8+4-1\choose4$ is what we desire, imagine that the * are mirrors, and | dividers, which determine which room the mirrors fall into. Since there are 8 rooms, we have 7 dividers, i.e., we seek the number of distinct combinations of

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