I received a word problem that goes like this.
A local kindergarten is thinking of making posters that show all the different ways of adding two or more integers from 1 to 9 to get a sum of 10. If there is enough space on each poster for up to 50 possible solutions, how many posters will the school need to make?
(Note: sums that contain the same number but in a different order are considered to be different; for example, 1 + 9 and 9 + 1 are two different solutions.)
What is the answer to this problem, and more importantly, how do I solve it?
To be clear, a solution such as,"1+1+1+1+1+1+1+1+1+1+1," is a valid one.
Please make your explanation somewhat thorough and be available to explain further. Thanks!
Since the summands must be at least $1$ and at most $9$, a particular solution to the question corresponds to the placement of one or more addition signs in the nine spaces between successive ones in a row of ten ones. $$1 \square 1 \square 1 \square 1 \square 1 \square 1 \square 1 \square 1 \square 1 \square 1$$ For instance, $$1 + 1 1 + 1 1 1 + 1 1 1 1$$ corresponds to the solution $1 + 2 + 3 + 4$. The number of such solutions is equal to the number of non-empty subsets of a nine-element set, which is $2^9 - 1$.
To see this, observe that we have two choices for each of the nine spaces, to place an addition sign or not place one. That gives us $2^9$ solutions. However, we cannot leave all the spaces empty since that would correspond to using the summand $10$, which is not permitted. Therefore, the number of admissible solutions is $2^9 - 1$.
This sort of problem is called a composition.
I will leave it to you to determine how many posters will be needed.