<s>[INST] Find the coefficient of $x^{70}$ in $(x^1-1)(x^2-2)(x^3-3)\cdots(x^{12}-12)$. 
Find the coefficient of $x^{70}$ in $(x^1-1)(x^2-2)(x^3-3)\cdots (x^{12}-12)$.

I tried to solve this problem using theory of equation the coefficient of $x^{70}$ will be the  sum of products taking two at a time. But this very very exhaustive I want to know some another method as it will be proficient in higher powers. 
 [/INST] Since $1+2+3+\ldots+12=78$, the term $x^{70}$ must arise from taking $x^k$ from $(x^k-k)$ for almost every $k\in\{1,2,\ldots,12\}$, except for some $j_1,j_2,\ldots,j_r\in\{1,2,\ldots,k\}$ such that $j_1<j_2<\ldots<j_r$ and $j_1+j_2+\ldots+j_r=8$.  There are very few such tuples $(j_1,j_2,\ldots,j_r)$:


*

*for $r=1$, $j_1=8$;

*for $r=2$, $(j_1,j_2)=(1,7),(2,6),(3,5)$;

*for $r=3$, $(j_1,j_2,j_3)=(1,2,5),(1,3,4)$.
Therefore, the coefficient of $x^{70}$ is
$$(-1)^1\cdot 8+(-1)^2\cdot (1\cdot 7+2\cdot 6+3\cdot 5)+(-1)^3\cdot(1\cdot 2\cdot 5+1\cdot 3\cdot 4)=4\,.$$
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