A friend of mine is giving a dinner party. His current wine supply includes 8 bottles of zinfandel, 9 of merlot, and 11 of cabernet (he only drinks red wine), all from different wineries. (a) If he wants to serve 3 bottles of zinfandel and serving order is important, how many ways are there to do this? (b) If 6 bottles of wine are to be randomly selected from the 28 for serving, how many ways are there to do this? (c) If 6 bottles are randomly selected, how many ways are there to obtain two bottles of each variety? (d) If 6 bottles are randomly selected, what is the probability that this results in two bottles of each variety being chosen? (Round your answer to three decimal places.) (e) If 6 bottles are randomly selected, what is the probability that all of them are the same variety? (Round your answer to three decimal places.)

A friend of mine is giving a dinner party. His current wine supply includes 8 bottles of zinfandel, 9 of merlot, and 11 of cabernet (he only drinks red wine), all from different wineries.

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

8 * 7 * 6 = 8! / (8 - 3)! = 336

(b) If 6 bottles of wine are to be randomly selected from the 28 for serving, how many ways are there to do this?

28 * 27 * 26 * 25 * 24 * 23 / 6! = 28! / (6! * (28 - 6)!) = 376740

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

8 * 7 / 2! * 9 * 8 / 2! * 11 * 10 / 2! = 55440

(d) If 6 bottles are randomly selected, what is the probability that this results in two bottles of each variety being chosen? (Round your answer to three decimal places.)

55440 / 376740 = 0.147

(e) If 6 bottles are randomly selected, what is the probability that all of them are the same variety? (Round your answer to three decimal places.)

8! / (6! * (8 - 6)!) + 9! / (6! * (9 - 6)!) + 11! / (6! * (11 - 6)!) = 574

574 / 376740 = 0.00152