How i write this problem in Mathematical Notation

Question

How many numbers can be created with the numbers $8,3,-15,-5,-10$ which sum is equal $-7$. And what is this pair of numbers?

My doubt is how to I represent this problem in mathematical notation and how to I find the solution.

I already make a computer algorithm to give me the answer:

8+-15=-7  
3+-10=-7  

I think it will help.

Answer 1

The way you write this problem is already correct. But if you like you can write in this way too:

Let $A:=\{8,3,-15,-5,-10\}$ and define the function

$$f:A\times A\to \Bbb Z,\quad (x,y)\mapsto x+y$$

How many solutions are for the equation $f(x,y)=-7$? List the solutions.

This is just an example, the problem can be stated in many other ways using some other formal notation.

Answer 2

$S=\{-15,-10,-5,3,8\}$

$A=\{(x,y)\in S^2|x+y=-7\}$

What is $|A|$? What is $A$?

Answer 3

Here is a different way to formulate the problem.

But first note that the problem as you state it does not restrict the number of summands. So, there is a also one more solution, namely adding up the following three quantities: \begin{align*} 8+(-5)+(-10)=-7 \end{align*}

There are no more than three solutions if we agree that each summand might occur at most once.

If you are already familiar with multiplication of polynomial expressions, we can encode the occurrence of zero or one $8$ as \begin{align*} 1+x^8 \end{align*} Similarly we can encode the occurrence of each of the other summands. Finding the result $-7$ could then be stated as

Problem: Find the coefficient of $x^{-7}$ in \begin{align*} (1+x^8)(1+x^3)(1+x^{-15})(1+x^{-5})(1+x^{-10}) \end{align*}