I am trying to figure out what this sum means:
$$\sum_{1 \le x <y\le5} 1$$
Up to now, I have only seen sums with a start and endpoint (start and end value) for the summation. For example:
$$\sum_{n=k}^{\infty}n$$
What does it mean if I have $1 \le x <y\le5$ as a lower bound of my sum?
EDIT: I have been trying to understand the answer that gt6989b has given but I don't arrive at the same answer. I basically have a double sum to evaluate:
$$\sum_{x=1}^4\sum_{y=x+1}^51=\sum_{x=1}^4(1+1+1+1)=\sum_{x=1}^4 4=16?$$
I am pretty sure I am doing something wrong but I am stuck. Can anyone point out my mistake?
Best Answer
$$ \sum_{1 \le x < y \le 5} 1 = \sum_{x=1}^4 \sum_{y={x+1}}^5 1 = 10 $$
In other words, you always have $x < y$ and $x \ge 1, y \le 5$ is enforced for every valid pair $(x,y)$. Hence, the shorthand expression is $$1 \le x < y \le 5$$ which tightly collapses all constraints into one multiple inequality.