Partial set of total combinations

Question

I am trying to figure out the formula for all combinations of a set, split into two sets.

For example, the total combinations for 10 choose (1 to 10) = 2^10 - 1 = 1023 combinations.

Is there a way to determine a partial set of the total combinations, as in:

10 choose (1 to 5)

without having to calculate the binomial coefficient 6 times in a row,

e.g. 10 choose 1 + 10 choose 2 + 10 choose 3 + 10 choose 4 + 10 choose 5

Answer 1

Only for the example you have shown, as that covers almost half of the expansion you can use this $\dfrac{2^{10} + {{10}\choose{5}}}{2} -1$.