Dimensional Dilemma!

Consider an $n$ -dimensional solid sphere of radius $R$ , with uniform charge-density and a total charge $Q$ . A charged particle $q$ is kept at a distance of $r$ from the center. For $r<R$ , what is the Net Coulombic Force experienced by the particle?

If it is in the form of

$\displaystyle\vec{F_{c}}= \left(\dfrac{\color{#3D99F6}{m}}{\color{#3D99F6}{p}.\varepsilon_{0}}\right).\left(\dfrac{(\color{#3D99F6}{a}+2)!}{\left(\Gamma\left(\dfrac{\color{#3D99F6}{b}}{\color{#3D99F6}{c}}\right)\right)^{\color{#3D99F6}{x}}.\color{#3D99F6}{w^{y}}}\right)$

Find $\color{#3D99F6}{ m+p+a+b+c+w+x+y}$ .

Details and Assumptions :

• $n=12$ , $Q=10 \text{ C}$ , $q=1.4 \text{ C}$ , $r=5 \text{ cm}$ , $R=13\text{ cm}$ .

• All the letters represent integers and are not necessarily distinct. $m$ and $n$ , $b$ and $c$ are coprime.

• $\varepsilon_{0}$ is the constant of permittivity of free space.