We have 8 bottles of Zinfandel. 10 of Merlot, and 12 of Cabernet.

If he wants to serve 3 bottles of zinfandel and serving order is important, how many ways are there to do this?

$\displaystyle P=8!/(8-3)! = 336$

If 6 bottles of wine are to be randomly selected for the 30 for serving, how many ways are there to do this?

$\displaystyle P=30!/6!(30-6)! = 593,775$

If 6 bottles are randomly selected, how many ways are there to obtain two bottles of each variety?

$\displaystyle P = {{\binom{8}{2}} {\binom{12}{2}} {\binom{10}{2}}} = 83,160

$

If 6 bottles are randomly selected, what is the probability that this results in two bottles of each variety being chosen?

$\displaystyle P = {{\binom{8}{2}} {\binom{12}{2}} {\binom{10}{2}}}/{\binom{30}{6}} = .1400

$

If 6 bottles are randomly selected, what is the probability that all of them are the same variety.

Not even sure how to start this one. Any help?

Are these right?