<s>[INST] product of terms taken $3$ at a time in polynomial expression 
Finding product of terms taken $3$ at a time in $\displaystyle \prod^{100}_{r=1}(x+r)$ 

Try: 
$$\displaystyle \prod^{100}_{r=1}(x+r)=x^{100}+(1+2+3+\cdots +100)x^{99}+(1\cdot 2+1\cdot 3+\cdots+100\cdot 99)x^{98}+(1\cdot 2\cdot 3+2\cdot 3 \cdot 4+\cdot\cdots+98\cdot 99\cdot 100)x^{98}+\cdots$$
for $1$ at a time (Coefficient of $x^{99}$) is $\displaystyle \sum^{100}_{i=1}i = 50\cdot 101$
for $2$ at a time (Coefficient of $x^{98}$) is $\displaystyle \sum^{100}_{i=1}\sum^{100}_{j=1\;, (1\leq i<j \leq 100)}i \cdot j= \frac{1}{2}\bigg[\bigg(\sum^{100}_{i=1}i\bigg)^2-\sum^{100}_{i=1}i^2\bigg]$
But could some help me how to find coefficient of $x^{97},$ thanks
 [/INST] With some manipulation (using inclusion exclusion principle) you can write the sum as:
$$\left(\sum_{i=1}^{100} i\right)^3 = \binom{3}{1}\left(\sum_{i=1}^{100} i^2\right)\left(\sum_{i=1}^{100} i\right) - 2\left(\sum_{i=1}^{100}i^3\right) + \binom{3}{1} \binom{2}{1}S $$
Where $S$ is the required sum.
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