How to compute for the number of sequential combinations possible in a set?

Question

Suppose I have a set {A, B, C, D, E, F, G}, and I need to find the number of possible subsets with N number of elements where each subset can only content neighboring elements.

Example: Find possible subsets where each contains 3 elements.

A     B     C     D     E     F     G
|           |     |     |     |     |
|____[1]____|     |     |     |     |
      |           |     |     |     |
      |____[2]____|     |     |     |
            |           |     |     |
            |____[3]____|     |     |
                  |     |     |     |
                  |____[4]____|     |
                        |           |
                        |____[5]____|

Here I have 5 subsets possible: [{A,B,C}, {B, C, D}, {C, D, E}, {D, E, F}, {E, F, G}].

Another example: Find possible subsets where each contains 2 elements

A     B     C     D     E
|     |     |     |     |
|_[1]_|     |     |     |
      |     |     |     |
      |_[2]_|     |     |
            |     |     |
            |_[3]_|     |
                  |     |
                  |_[4]_|

Here I have 4 subsets possible: [{A, B}, {B, C}, {C, D}, {D, E}].

How can this be achieved mathematically?

Answer 1

Assume that $i$ is the first of the elements in such a choise. As the rest needs to be neighbors, they have already been fixed as $i+1, i+2,...,i+(N-1)$. Thus the only choise we can make when choosing a premutation is to choose the smallest part. This may be done in $K-(N-1)$ ways, where $K$ are the number of elements in your set and $N$ are the number of elements you should choose. We need to subtract $N-1$ as the last $N-1$ elements may not be the first element if we need to choose N elements.