Did I solve this probability problem correctly?
The question goes like this:
In every table in Earl’s Diner, there are exactly four pairs of chopsticks, and each pair is uniquely colored. If two random customers pick four chopsticks at random, what is the probability that they pick one for each color?
The answer is in the form $\alpha/\beta$ for some coprime integers $\alpha$ and $\beta$. Determine the value of $\alpha + \beta$.
My approach went like this: there are four times that the customers will pick a chopstick, and since there are $2$ chopsticks for every color, then by FCP the number of ways that they will pick one of each color will be $$ 2 \times 2 \times 2 \times 2 = 16 $$ And then the total number of ways you can select four chopsticks would be $8C4$, which would be equivalent to $70$.
Then the probability would be $16:70$ or $8:35$ giving the answer of $43$. I am unsure if this is the correct solution and answer, can someone please verify it?