Answer
$\displaystyle \frac{2}{21}$
Work Step by Step
$P(E)=\displaystyle \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}=\frac{n(E)}{n(S)}$
$n(S)=C(10,5)=252$
$E$ = ($2$ of 3 greens) AND $(1$ of 4 reds) AND ($1$ of 2 whites) AND ($1$ of 1 purple)
$n(E)=C(3,2)\cdot C(4,1)\cdot C(2,1)\cdot C(1,1)=3\cdot 4\cdot 2\cdot 1=24$
$P(E)=\displaystyle \frac{24}{252}=\frac{2}{21}$