# Scary Sums and Products!

Number Theory Level 4

Given that $f(x)=x^4+10 x^3+35 x^2+50 x+24$ and that $S=\prod_{i=1}^{50} \left( \prod_{j=1}^{50} (f(i)-f(j)) \right),$ find the value of $\left(\frac{S \cdot (\sum_{k=1}^{f(50)} f(k))}{\sum_{k=1}^{f(49)}f(k)}+f(1) \right) \pmod{1000}$

×