Notation $\sum_{k \neq k'}$: is the order important?

Question

I am reading The Elements of Statistical Learning.

At pag. 309 the Gini index is defined as:

$$ \text{Gini} = \sum_{k \neq k'} \hat{p}_{mk} \hat{p}_{mk'} $$

where the index $k = {1,2,\dots,K}$.

In case $K=2$, does the notation implies

$$ \text{Gini} = \hat{p}_{m1} \hat{p}_{m2} $$

Or is it

$$ \text{Gini} = \hat{p}_{m1} \hat{p}_{m2} + \hat{p}_{m2} \hat{p}_{m1} = 2 \hat{p}_{m1} \hat{p}_{m2}$$

Is the order important?

Is this kind of notation "officially" defined? Where is it used?

Answer 1

$$ \sum_{k \neq k'} \hat{p}_{mk} \hat{p}_{mk'}= \hat{p}_{m1} \hat{p}_{m2} + \hat{p}_{m2} \hat{p}_{m1}. $$

$$ \sum_{k \lt k'} \hat{p}_{mk} \hat{p}_{mk'}= \hat{p}_{m1} \hat{p}_{m2}.$$