a) Nope.

To get 0, you need to have exactly 10 of -1 and 10 of 1.

Now the number of sequences is

20! is 'shuffling' all the 20 terms

10! is dividing by the number of ways -1 can be arranged because there are 10 similar -1

The other 10! is for 1.

EDIT: Cross that out, I messed up P and C... again -.-

b) I don't know the shortcut (though I'd like to learn about it) but I could do by the long way.

For the first k numbers, the sum is positive when you have:

k = 1, first term 1

k = 2, first terms 1 1

k = 3, first terms 1 1 1, or 1 1 -1

k = 4, first terms 1 1 1 1, or 1 1 1 -1

k = 5, first terms 1 1 1 1 1, or 1 1 1 1 -1, or 1 1 1 -1 -1

.

.

.

k = 10, first terms 10(1), or 9(1)1(-1), or 8(1)2(-1), or 7(1)3(-1), or 6(1)4(-1)

Then, you multiply by 2 due to the symmetry for the last term to be negative.

When k = 1, you have

When k = 10, you have

Okay, maybe a summation formula would simplify that calculation...