Q:

How many $4$-subsets of $S= \{1,2, \ldots,20\}$ contain no consecutive integers? 

I am trying an assignment problem  which I am unable to think about.  

Question is -> How many $4$-subsets of $S= \{1,2, \ldots,20\}$ contain no consecutive integers?  

I don't have any idea how to solve it.  
Can someone please explain how to think about it ?  

A:

Consider any subset $(A,B,C,D)$ with $A>B>C>D$.
Define
$x_1 = 21-A$
$x_2 = A - 1 - B$
$x_3 = B - 1 - C$
$x_4 = C - 1 - D$
$x_5 = D$
$x_1 , x_2 , ... , x_5$ are all positive integer greater than or equal to $1$ and
$x_1 + x_2 + x_3 +x_4 + x_5 = 18$.
For every $(A,B,C,D)$ there is a unique $(x_1 , x_2 , x_3 , x_4 , x_5)$.
There are ${{18-1} \choose {5-1}}$ possible $(x_1 , x_2 , x_3 , x_4 , x_5)$, thus there are ${17 \choose 4}$ such 4-subsets.