The Binomial expansion of (1 + x)^n is,
1 + nx + n(n-1).x^2/2 + n(n-1)(n-2).x^3/3! + ...
with the kth term given as tk = n(n-1)(n-2) ... (n - (k-2)).x^(k-1)/(k-1)!, k > 2
We have (2x-4)^21, which becomes
(-4)^(21)(1 - (x/2))^(21)
(-4)^(21)(1 + (-x/2))^(21)
The expansion of this is,
-4^(21){1 + 21(-x/2) + 21*20*(-x/2)^2/2 + 21*20*19*(-x/2)^3/3! + ...
And the 13th term (k=13) is,
-4^(21){21*20*19* .... *12*11*10}*(-x/2)^(12)/12!}
-4^(21)*146,965x^(12)/2048
-2*4^(15)*146,965x^(12)
-293,930*4^(15)*x^(12)
Answer: 13th term in the expansion is: -293,930*4^(15)*x^(12)