math challenge question consecutive integers subsets?

If the subset is 2 integers, there are 9 consecutive subsets
12, 23, ... 910
If the subset is 3 integers, there are 8 subsets
123, 234, ... 8910
etc,
The total number is 9 + 8 + ...  + 1 = 9*10/2 = 45

Another way to count them: for any pair of integers, there is one subset of consecutive integers from the smallest to the largest of the pair.

The number of pairs is the same as the number of ways to select 2 numbers from 10 without replacement, which is 10x9/2 = 45.

A third way:
There are 9 possible subsets starting with 1 (12, 123, 1234, ... 12345678910)
8 possible subsets starting with 2 (23, 234, ... 2345678910
...
1 subset starting with 9