One of Each Colour

In every table in Robert's Diner, there are exactly four pairs of chopsticks, and each pair is uniquely colored. If a customer randomly pick four chopsticks, what is the probability that he/she picks one for each color? The answer is in the form $\frac{\alpha}{\beta}$ for some coprime integers $\alpha$ and $\beta$. Input the value of $\alpha+\beta$. Details and assumptions: • there are exactly four colors available, i.e. each pair has the same color • the chopsticks are not packed in pairs • each chopstick is equally likely to be picked