Question

What is the second term in the binomial expansion of (2r-3s)12?

Answer 1

$(a+b)^n=\displaystyle\sum_{k=0}^n\binom nka^{n-k}b^k$

where $\dbinom nk=\dfrac{n!}{k!(n-k)!}$. The second term of the expansion occurs when $k=1$.

So the second term of the expansion of $(2r-3s)^{12}$ is

$\dbinom{12}1(2r)^{12-1}(-3s)^1=12(2r)^{11}(-3s)=-73728r^{11}s$