Expected Number of Distinct Colors When Picking Balls

Twenty five colored balls are placed in a box: $5$ red, $5$ blue, $5$ yellow, $5$ green, and $5$ orange. What is the expected number of distinct colors in $3$ randomly picked balls? Assume that the balls are equally likely to be picked. The answer is in the form $\frac{\alpha}{\beta}$ for coprime integers $\alpha$ and $\beta$. Input the answer as the sum $\alpha +\beta$.