Answer
$\displaystyle \frac{11}{12}$
Work Step by Step
$P(E)=\displaystyle \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}=\frac{n(E)}{n(S)}$
$n(S)=C(10,5)=252$
$E$ = not all green.
$E'$ = (all 3 of 3 greens) AND (2 from the remaining 7)
$n(E')=C(3,3)\cdot C(7,2)=21$
$P(E')=\displaystyle \frac{21}{252}=\frac{1}{12}$
$P(E)=1-P(E')=\displaystyle \frac{11}{12}$