Fifteen cards

Can you use the information to find out which cards I have used?

Problem

Fifteen Cards printable sheet

I have fifteen cards numbered $1 - 15$.

I put down seven of them on the table in a row.

Image

 

The numbers on the first two cards add to $15$.

The numbers on the second and third cards add to $20$.

The numbers on the third and fourth cards add to $23$.

The numbers on the fourth and fifth cards add to $16$.

The numbers on the fifth and sixth cards add to $18$.

The numbers on the sixth and seventh cards add to $21$.

What are my cards?

Can you find any other solutions?

 

How do you know you've found all the different solutions?