TITLE: How to answer the following combinatorial question?
QUESTION [5 upvotes]: Consider the set $S = \{ 1,2, \cdots , 15 \}$. The general question is: how many subsets of $S$ exist such that the subsets contain 4 different elements without any consecutive numbers? 
As a sub-question, I have to show that this problem is equivalent with finding the number of solutions $$(*) \qquad a_1 + a_2 + a_3 + a_4 + a_5 = 14 ; \qquad a_1,a_5 \geq 0 ; a_2, a_3, a_4 \geq 2 $$
To prove this, I tried the following. Consider the numbers 1 through 15 laid out next to each other in a row, starting with the number 1 and ending with 15. To find the subsets that obbey the given restrictions, we start of by picking some number $a_1 \geq 1$. Then, $a_2$ cannot be a consecutive number, so the distance to $a_1$ should be larger than or equal to two. Similar for $a_3$ and $a_4$. Then I don't know about $a_{5}$. 
However: one is asked to prove the original problem is equivalent to proving that $a_1 \geq 0$, not $a_1 \geq 1$. Furthermore, I don't understand why $a_{5}$ enters the scene here: we should find subsets containing $4$ elements, right? Could you please explain this? 
Furthermore, I don't understand why the $(*)$-marked sum should be equal to 14. Could you please this explain this as well, or give me a hint? 
Furthermore, I am asked to show that this number of subsets is equal to the coefficient of $x^{14}$ of the generating function $$f(x) = \frac{x^6}{(1-x)^5} .$$
How does one show that? Or could you please give a hint?

REPLY [3 votes]: Use 'Stars and Bars' to count.
We can represent the four numbers as a string of 4 'bars' and 11 'stars', where the position of each the bar counts off a number. 
So the string: "$**|*|***|**|***$" represents: $\{3, 5, 9, 12\}$.
Since we need at least one 'star' between each pair of 'bars', this is equivalent to finding the permutations of 4 bars and 8 remaining stars.
So there are $\dbinom{12}{4} = \dfrac{12!}{4!\;8!} = 495$ ways to do this.

Your next step is to show that the problem of five variables can also be represented by the same string.