Coefficient of $x^{-2}$ in expression $(x-1/x)^{12}$

Question

Find the coefficient of $x^{-2}$ in $$(x-1/x)^{12}$$

Can anyone help me with this by providing the process to find the answer? I know the answer is $-792$ but I can't find it.

Answer 1

You use $$(a+b)^n=\sum_{k=0}^n \binom nk a^kb^{n-k}$$ with $a=x,\ b=-\dfrac1x,\ n=12$. It becomes $$\left(x-\frac1x\right)^{12}=\sum_{k=0}^{12} \binom {12}k (-1)^{12-k} x^{2k-12}$$

Now look for $-2$ among all those exponents of $x$.

Answer 2

$$(x-1/x)^{12}=x^{-12}(x^2-1)^{12}=\sum_{k=0}^{12}\binom{12}{k}x^{2k-12}(-1)^{12-k}$$ $x^{-12+2k}=x^{-2}$ so $k=7$ that mean desired coefficient is $$\binom{12}{7}(-1)^{5}=-\binom{12}{7}$$