Combinations problem

Question

I'm in the last year of high school, but my school's math program is really bad and I'm trying to catch up on my math. I've been staring at this problem trying to find an efficient method to solve it rather than brute forcing every alternative, but I can't find one. I'd love some help please!

Mum cut 8 roses from her garden. She had two vases - one tall and the other shallow. If she put at least one rose in each vase, how many different combinations of roses could she have in the two vases altogether?

Answer 1

If the roses are indistinguishable, a combination is completely determined by the number of roses in each vase. Consider the tall vase: it can have $1,2,3,4,5,6$, or $7$ roses, so there are just seven distinct combinations.

If the eight roses can be distinguished from one another, matters are very different. Now we care not only how many roses are in each vase, but which roses. Suppose that you’re making an arrangement: you go through the $8$ roses one by one, and for each one you decide whether to put it in the tall vase or in the shallow one. That’s a sequence of eight two-way choice; how many ways are there to make such a sequence? Be careful, though: that counts every possible arrangement, including those that have all eight roses in the same vase. How many of these ‘bad’ arrangements are there that have to be subtracted from the preliminary total?

Answer 2

For a moment, forget about the constraint that at least one rose is in each vase.

If the roses are considered to be indistinguishable, there are 9 possibilities: any number of roses may be in the tall vase (0 through 8) and the rest are determined to be in the short vase.

If the roses are considered to be distinguishable, then the answer is $2^8 = 256$, because this is the number of subsets of roses that are possible for the tall vase; again which roses are in the short vase is determined.

The constraint excludes two particular configurations, that in which all roses are placed in the tall vase and that in which all roses are placed in the short vase, both of which are counted in both interpretations above. So the answers for each interpretation respectively are $7 = 9 - 2$ and $254 = 256 - 2$.