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 x^{1}) is just the sum of the factors. Meaning: (x-1)(x-2) = x^{2} - 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) = x^{2}-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

(x^{2} - 3x + 2)(x^{2}-7x+12) = x^{4} - 10x^{3} +35x^{2} - 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

(x^{4} - 10x^{3} + 35x^{2} - 50x +24)(x^{4} - 26x^{3} + 485x^{2} -18930x+168) =

x^{8} - 36x^{7} + ....

(-1)+(-2)+(-3)+(-4)+(-5)+(-6)+(-7)+(-8)=-36

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)