Creating a $6$-integer subset without repetition with no consecutive integers from $\{1,2,\dots,20\}$

Question

How many subsets of six integers chosen without repetition from $1,2,\dots, 20$ are there with no consecutive integers?

I am unsure of where to begin with this problem.

Answer 1

Lets choose $6$ elements $p_1,p_2,p_3,p_4,p_5,p_6$ from $\{1,2,\cdots, 15\}$, such that $p_i<p_j$ for $i<j$.

There are $15 \choose 6$ ways to do so.

Now, for our problem statement, lets inject a gap of $1$ between $p_i, p_{i+1}$. This will create a subset of numbers from a set containing $20$ elements. However, there is one to one isomorphism from subset of $15$ elements, and subset of $20$ elements (where the later has an injected gap).

So answer is $15 \choose 6$, which is 5005.