Question: Convert "Sum of the products of the ten numbers +-(1,2,3,4,5) taking two at a time " in sigma notation format. 

I know many will think me stupid asking such type questions but please note i am beginner to pre-calculus so please don't close my question.

Answer: 
If you are allowed to use double sums, then

$$\sum\_{i = -5}^{5}\sum\_{j = -5}^{5} ij $$

is a sum of all possible products of the numbers $\{-5, ... , 5\}.$

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If you want you could exclude the cases $i,j = 0$, do like so:

$$\sum\_{i = -5 \\ i \neq 0}^{5}\sum\_{j = -5 \\ j \neq 0}^{5} ij, $$

but I would argue that this is less readable. Note that the case with $i,j = 0$ does't add anything to the sum, so the two sums above are equal.

Also note that this sum treats the pairs $(-2,3)$ and $(2,-3)$ distinctly, so the sum will look like

$$... + (-2)(3) + ... + 3(-2) + ... $$

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I also could imagine using the following notation: Let $P$ be the set of numbers $-5$ to $5$, excluding $0$; then your sum can be written as follows:

$$\sum\_{i,j \in P} ij. $$

These are all the same, that is, they all mean the same thing. Pick the one you like, there is no hard rule on what is "best". To me, often, making the sum readable is valued.
You can also say
$$\sum\_{(-5 \le i, j \le 5)}{ij}$$
This basically says, $i, j$ are integers between $-5$ and $5$. Take every possible combination of them and add the products.