how many subsets with 4-elements?

Question

I ran into a non-trivial question on one Statistics Book,

how many subsets with 4-elements from set {$1,2,...,10$} can be taken such that hasn't two consecutive number in that subsets.

how we should be able to solve this?

Answer 1

Consider a binary vector of size $10$, with exactly $4$ ones, such that if the $i$th bit is $1$ we take $i$ to our subset.

Now since we want that there wont be any $2$ consecutive ones, we'll start with $4$ ones and put between them one $0$, now we have $3$ more zeroes (since we spent $3$ for the gaps between our ones), and we can put them in those 3 gaps and as well before the first one and after the last one, to do that we have $$CC_5^3={{3+5-1}\choose {3}}$$

Answer 2

Here is an alternate way to think of this:

Put 6 dots in a line, representing the 6 numbers not chosen.

Then put a line in 4 out of the 7 gaps, representing the 4 numbers which are chosen;

there are $\dbinom{7}{4}$ ways to do this.