If $\frac{2 x^2+5 x+6}{(x+2)^3}=\frac{a}{x+2}+\frac{b}{(x+2)^2}+\frac{c}{(x+2)^3}$ then $a \cdot b+b \cdot c+c \cdot a=$

If $\frac{x^3 - 6x^2 + 10x - 2}{x^2 - 5x + 6} = f(x) + \frac{A}{x - 2} + \frac{B}{x - 3}$,then $f(x) = $

If $\frac{x-2}{x^2(2x-3)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{2x-3}$, then $2(A-C) = $

If $\frac{2x^3+1}{2x^2-x-6} = ax+b+\frac{A}{px-2}+\frac{B}{2x+q}$,then $51apB=$ (in $bqA$)

If $\frac{6 x^3+7 x^2+6 x-3}{(x-1)(x+3)\left(x^2+1\right)}=\frac{A}{x-1}+\frac{B}{x+3}+\frac{C x+D}{x^2+1}$ and $n=A+B+C+D$ and ${ }^{50} C_n={ }^{50} C_r$,then $r$ is equal to

For $a > 0$,let $\frac{1}{a(a+1)(a+2) \ldots(a+20)}=\sum_{k=0}^{20} \frac{A_{k}}{a+k}$. Then the value of $100\left(\frac{A_{14}+A_{15}}{A_{13}}\right)^{2}$ is equal to $....$