# Counting all subsequences constrained to a condition

I was trying to find all subsequences constrained to the following conditions:

1. remove element from the end.
2. remove element from the beginning.
3. remove element from both sides.

For example, given the sequence $$(1,2,3,4)$$ have subsequences $$\{(1,2,3,4), (1,2,3), (1,2), (1), (2,3,4), (2,3), (2), (3,4), (3), (4)\}$$. So, we have get total 10 subsequences.

Attempt: I was thinking of using combinators $${N}\choose{i}$$, where $$N$$ is the length of subsequence in our case and $$i$$ is the limit boundary of subsequence that starts at $$i=1$$ and increase all the way to $$N-1$$. However, since jumps are not allowed, this approach does not seem valid as it will count subsequence like $$(1,3,4)$$ which is not allowed.

Can you think of an hint/approach based on combinators/permutations to solve this please if possible or this can only be solved iteratively (through for loops/recursion)?

If I understand it correctly, you are counting the number of continuous subsequences. Every continuous subsequence one-to-one corresponds to a pair of positions representing its start position and end position, so the total number is $$\binom{N}{2}$$.