a
Volume of removed sphere

\({V_{remo}} = \frac{4}{3}\pi {\left( {\frac{R}{2}} \right)^3} = \frac{4}{3}\pi {R^3}\left( {\frac{1}{8}} \right)\)

Volume of the sphere (remaining)

\({V_{remain}} = \frac{4}{3}\pi {R^3} - \frac{4}{3}\pi {R^3}\left( {\frac{1}{8}} \right)\)

\( = \frac{4}{3}\pi {R^3}\left( {\frac{7}{8}} \right)\)

Therefore mass of sphere carved and remaining sphere are at respectively \(\frac{1}{8}\,M\,and\,\frac{7}{8}\,M.\)

Therefore, gravitational force between these two sphere 

\(F = \frac{{GMm}}{{{r^2}}}\)

\( = \frac{{G\frac{{7M}}{8} \times \frac{1}{8}M}}{{{{\left( {3R} \right)}^2}}} = \frac{7}{{64 \times 9}}\frac{{G{M^2}}}{{{R^2}}}\)

\( = \frac{{41}}{{3600}}\frac{{G{M^2}}}{{{R^2}}}\)