Drilling a sphere

Let $r$ and $R$ be the radius of the cylinder and sphere respectively. The volume of the remaining sphere is equal to the volume of the sphere minus the volume of the cylinder and volume of the two spherical caps. From the diagram above,

$r^2=R^2-5^2$

$V_{cylinder}=\pi r^2h=\pi (R^2-25)(10)=10\pi R^2-250\pi$

$V_{spherical~caps}=\dfrac{\pi y^2}{3}(3R-y)(2)=\dfrac{2\pi}{3}(R-5)^2[3R-(R-5)]=\dfrac{2}{3}\pi(R^2-10R+25)(3R-R+5)=\dfrac{2}{3}\pi(2R^3-20R^2+50R+5R^2-50R+125)=\dfrac{4}{3}\pi R^3-10\pi R^2+\dfrac{250}{3}\pi$

$V_{sphere}=\dfrac{4}{3}\pi R^3$

$V_{remaining}=\dfrac{4}{3} \pi R^3-(10\pi R^2-250\pi)-\left(\dfrac{4}{3}\pi R^3-10\pi R^2+\dfrac{250}{3}\pi\right)=\dfrac{4}{3} \pi R^3-10\pi R^2+250\pi-\dfrac{4}{3}\pi R^3+10\pi R^2 - \dfrac{250}{3}\pi=$ $\boxed{\dfrac{500}{3}\pi}$