Define t_k (disregarding the coefficient) in the binomial expansion of (x

-1

2

+x

-1

3

)ⁿ as follows:

t_k=(x

-1

2

)^n-k+1(x

-1

3

)^k-1=x

k-3n-1

6

=x

-17

6

So k-3n-1=-17, and k=3n-16. We find that the only lattice points satisfying both k=3n-16 and k
k=2 and n=6
k=5 and n=7
k=8 and n=8

The binomial coefficient of these terms is the number of combinations possible with n objects taken k-1 at a time - or, respectively, 6, 35, and 8.

We can also solve this problem using arbitrary exponents a and b, provided that a and b are whole numbers (non-negative integers):

(x

-1

2

)^a(x

-1

3

)^b=x

-3a-2b

6

=x

-17

6

3a+2b=17

17-3a must be an even number, so a must be odd. Therefore, we can have a=1 and b=7, a=3 and b=4, or a=5 and b=1, and the binomial coefficient will be the number of ways that a+b be objects can be taken a at a time - 8, 35, and 6.