How many are there?

mathVNpro

In the set of $n$ positive interger numbers $S=\{1,2,...,n\}$ . $(a,b,c,d)$ is $4$ numbers which is taken from $S$ . $(a,b,c,d)$ is call good if we have $a+b=c+d$ . How many good $(a,b,c,d)$ ?

**Posted:** Sun Nov 08, 2009 2:53 pm

evoluciona · New Member Offline Joined: 27 Oct 2009 Posts: 6

Let f(m) = the number of solutions of a+b=m with a,b in S.
Then you're after
$\sum_{m=2}^{2n} f(m)^2.$

f(m) takes the values 1,2,3,...,n,...,3,2,1 on this sequence of m - might have this wrong Smile

so you want

$2\sum_{j=1}^{n-1} j^2 + n^2 = (n-1)n(2n-1)/3 + n^2 = 2n^3/3 + n/3$
-evo

**Posted:** Sun Nov 08, 2009 4:42 pm

mathVNpro

**Posted:** Sun Nov 08, 2009 4:46 pm