$\displaystyle 8 $ bottles of zinfandel

$\displaystyle 10 $ bottles of merlot

$\displaystyle 12 $ bottles of cabaret.

(a) If he wants to serve $\displaystyle 3 $ bottles of zinfandel and serving order is important, how many ways are there to do this?

$\displaystyle \frac{8!}{5!} = 336 $

(b) If $\displaystyle 6 $ bottles of wine are to be randomly selected from the $\displaystyle 30 $, how many ways are there to do this?

$\displaystyle \binom{30}{6} = 593,775 $

(c) If $\displaystyle 6 $ bottles are randomly chosen, how many ways are there two bottles of each variety being chosen?

(d) If $\displaystyle 6 $ bottles are randomly chosen, what is the probability that this results in two bottles of each variety being chosen.

(e) If $\displaystyle 6 $ bottles are randomly chosen, what is the probability that they are all the same variety?