I don't know if there is a faster way to solve this problem, but here is I how I solved it:

Divide into cases:

First two numbers are 1 and 2.

There are three possible numbers for the third number (3,4,5). We have 3 cases here.

First two numbers are 1 and 3:
There are two possible numbers for the third number(4,5). We have 2 cases here.

First two numbers are 1 and 4:
Just one possible number for the third number(5). 1 case

First two numbers are 2 and 3:
Two possible numbers for the third number (4,5). 2 cases

First two numbers are 2 and 4:
Just one possible number for the third number (5). 1 case

The last case is the first is formed by the sequence 3-4-5.

We have 3+2+1+2+1+1=10 possibilities. Answer is C