Question

If nc4 , nc5 and nc5 are in AP then the value of n can be

Answer 1

If both nc5 and nc5 are in the sequence, then d=0. I shall assume a typo, and that you meant nc6.
So, now you have
nc5 - nc4 = nc6 - nc5
Expanding this out gives you a 5th-degree polynomial equation
(n-14)(n-7)(n-3)(n-2)(n-1) = 0
Since n has to be at least 6, that means n=7 or n=14.
Check:
n=7 gives: 35,21,7
n=14 gives: 1001,2002,3003