Find The Sum

Algebra Level 3

If 13221!+23322!+33423!++10031012100!=1ab \dfrac{1^3 - 2^2}{1!} + \dfrac{2^3 - 3^2}{2!} + \dfrac{3^3 - 4^2}{3!} + \cdots + \dfrac{100^3 - 101^2}{100!} = 1 - \dfrac{a}{b} for some coprime positive integers a,b,a, b, find the last three digits of a.a.

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