Find the number of ways in which three numbers can be selected from the set $\{1,2,\dots,4n\}$ such that the sum of the three selected numbers is divisible by 4.
My approach: If $n=1$ then the set is $\{1,2,3,4\}$ and only one valid combination exists, $\{1,3,4\}$. But in general how can I find the solution? Please help.
If three natural numbers add to a multiple of four, there are five possibilities for the individual numbers' residues modulo 4 up to ordering: $$000, 112, 220, 332, 013$$ Within $\{1,2,\dots,4n\}$ each residue modulo 4 occurs $n$ times. The number of valid selections for the $000$ case (i.e. each number in the selection is $\equiv0\bmod4$) is $\binom n3$, that for each of the $112/220/332$ cases $n\binom n2$ and that for the $013$ case $n^3$, so the total number of ways is $$\binom n3+3n\binom n2+n^3=\frac{8n^3-6n^2+n}3$$