Question

Given a, b, c, d (where a 13 years ago

Answer 1

From my understanding, this could also be phrased as:

How many pairs \((a,d)\) such that \(1\leq a,d\leq2012\), \(a<d\), and \(3|(d-a)\).

Answer 2

So if \(a=1,2\) then you would have 670 choices for \(d\). If \(a=3,4,5\) then you have \(669\) choices for \(d\). And so on. That means I believe your final answer should be\[\large 2(670)+3\sum_{n=1}^{669}n\]