Your cards could be:
*9,6,14,9,7,11,10
*10,5,15,8,8,10,11
*8,7,13,10,6,12,9
*7,8,12,11,5,13,8
*6,9,11,12,4,14,7
or
*5,10,10,13,3,15,6
I worked this out by writing a list of what the numbers could be.
The first two numbers had to add up to fifteen, so the first two numbers could only be something from one to fourteen. Fifteen would mean the other number would be zero but that can't be true because the cards are only labelled one to fifteen. This means there is fourteen options.
The second and third cards have to add up to twenty so I subtracted the second card from twenty to get the third card. But, some of those cards' numbers are over fifteen so we need to eliminate some options. By now, you might see the pattern of counting down and up in the diagram (uploaded).
The third and fourth numbers need to add up to twenty three, so I used the same strategy of subtracting the third number from twenty-three to get the fourth number. Again, some of these numbers are over fifteen so we need to eliminate some more options. Now we have eight options.
Let's see if there is any numbers over fifteen when we use that same strategy to find the fifth, sixth, and seventh numbers! Four and five's numbers need to add up to sixteen, five and six eighteen, and six and seven twenty-one.
As you can see, there is some more numbers over fifteen so we need to eliminate some more options. This leaves you with six options.
That was my first solution. Someone challenged me to look again at the answers and I was met by a mistake. If there was fifteen cards labelled one to fifteen, each one had to be labelled differently. so why was one of my answers 10,5,15,8,8,10,11? I had to eliminate four more options. That left me with two final options:
*8,7,13,10,6,12,9
and
*6,9,11,12,4,14,7
I know I have found all the options because the way I set out and solved the problem covers all possibilities.