starting from the rules,I will refer to these in my answers. they are:
The 1st two cards add up to 15,
the 2nd card and the 3rd card add up to 20,
the 3rd & the 4th card adds up to 23,
the 4th & 5th card adds up to 16,
the 5th & 6th card adds up to 18,
and the 6th & 7th card adds up to 21.
there are only 2 possible answers and here is evidence and the wrong answers with the reasons why :
1) 1,14,6(wrong because you need to add 17 to 6 to make 23 and you only have numbers up to 15)
2) 2,13,7(wrong because you need to add 16 to 7 to make 23 and the numbers only go up to 15)
3) 3,12,8,15,1(wrong because there is too much to add to 1 to make 18)
4) 4,11,9,14,2(wrong, to much to add to 2 to make 18)
5) 5,10,13,7,9(wrong,because next you need two nines)
6)* 6,9,11,12,4,14,7 (*this is full and correct)
7) 7,8,12,11,5,13(wrong,you need another 8)
8)* 8,7,13,10,6,12,9(*this is full,correct and is the last answer although i'll show you the errors of the rest of the combinations.)
9) 9,6,14(wrong, needed another 9 to complete)
10) 10,5,15,8(wrong,needed another 8 to complete)
(the rest of these have the same error, there was too much to add to make 23. I have said that just in case you wonder why there is no definition.)
11) 11,4 (wrong)
12) 12,3(wrong)
13) 13,2(wrong)
14) 14,1(wrong)
so there is my solution and good luck on the other challenges!