\(A=\dfrac{n!+2}{\dfrac{n!}{\left(n-k\right)!}\cdot n!-k}+\dfrac{3003+10010+6435}{19448}\)
\(=\dfrac{n!+2}{n\left(n-1\right)\cdot...\cdot\left(n-k+1\right)\cdot n!-k}+1=\dfrac{n!+2+\dfrac{n!^2}{\left(n-k\right)!}-k}{\dfrac{n!^2}{\left(n-k\right)!}-k}\)
\(B=\dfrac{n!-\left(n-1\right)!}{\left(n-2\right)!}=\dfrac{\left(n-1\right)!\left(n-1\right)}{\left(n-2\right)!}=\left(n-1\right)^2=n^2-2n+1\)