summation - Trouble understanding $sum$ notation

Question:

What does

$$\left(\sum_{a, b, c}a\right)^2$$ mean ?

The answer given is $(a+b+c)^2$. However, I am having trouble understanding this.

I have seen this, this and this. But none helped.

Please help me understand this.

EDIT:

Also, what would be a better way to write:

$$\sum_{a,b,c}(b-c)(b+c)$$

Thanks.

Answer

The notation is very bad, since the summation index is apparently named $a$, while $a$ is also one of the values it takes. Better notations for this would be

$$\left(\sum_{k=a,b,c} k\right)^2\quad\text{or}\quad\left(\sum_{k\in\{a,b,c\}} k\right)^2.$$
Here the summation index is explicitly labeled $k$ and it takes values $a$, $b$ and $c$, so the sum is $a+b+c$.

The sum in the edit is even worse, I can't tell you what it means. I'd say it is just wrong. Summations should come with a summation index, always.

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In the comments you mentioned that they call these summations cyclic expression, which does indeed help at guessing what it's supposed to be. Consider the summand $(b-c)(b+c)$ as an expression in $a,b,c$ and then look at the same term for the cyclic permutations $b,c,a$ and $c,a,b$. So I guess they want

$$\sum_{a,b,c}(b-c)(b+c) = (b-c)(b+c) + (c-a)(c+a) + (a-b)(a+b).$$