Consider the problem :

Given $\displaystyle S=\left\{1,2,\cdots, 15\right\}$. How many subsets S' of 4 different elements can we make, such S' contain no consecutive numbers?

I like to show that this problem is equivalent with finding the number of solutions of:

$\displaystyle a_1+a_2+a_3+a_4+a_5=14; \ a_1,a_5\geq 0; \ a_2,a_3,a_4\geq 2 $

I'm not seeing any connection yet. A little insight please?