I am not sure how to do this with the Binomial Theorem, but I started to do the multiplcations.
First I did it in pairs, so (x-1)(x-2) and multiplied (x-3)(x-4)....
If n = the degree (the highest power), then the coefficient of the (n-1)th term (here the x1) is just the sum of the factors. Meaning: (x-1)(x-2) = x2 - 3x + 2 Notice, the (n-1)th term is just the (-1) of the first factor + (-2) of the second factor.
Do it again with (x-3)(x-4) = x2-7x+12 Notice, the (n-1)th term has a coefficient of (-3) + (-4).
In the next layer, take these quadratics in pairs, and get
(x2 - 3x + 2)(x2-7x+12) = x4 - 10x3 +35x2 - 50x + 24
Here, the (n-1)th coefficient is -10. Notice that this = (-1) + (-2) + (-3) + (-4) <-- the sum of the factors involved)
I did one more layer, this time using the first 8 factors, to find that
(x4 - 10x3 + 35x2 - 50x +24)(x4 - 26x3 + 485x2 -18930x+168) =
x8 - 36x7 + ....
Again, notice that the (n-1)th coefficient is the sum of all of the first n factors (8 factors in this case)
Notice that the highest power when we get finished multiplying all of these will be n = 50, so the (n-1)th term will be the 49th term. This is the coefficient you are looking for.
So, if you add (-1)+(-2)+(-3)+(-4)+ ... +(-47)+(-48)+(-49)+(-50) = -1275
Answer is (B)