Solutions - 5070-06

This is a standard multinomial coefficient application.

Number the marbles 1, 2, … 10 and then count the ways of putting them into three bins: red bin size 2, white bin size 3, blue bin size 5.

In other words, for the three bins: $r_1=2$, $r_2=3$, and $r_3=5$.

$$\begin{array}{c}\left(\genfrac{}{}{0px}{}{n}{r_1,r_2,r_3}\right)\gg \gg \left(\genfrac{}{}{0px}{}{10}{\mathrm{2,3,5}}\right)\\ \\ \gg \gg \frac{10!}{2!3!5!}\gg \gg 2520\end{array}$$