A friend of mine is giving a dinner party. His current wine supply includes 11 b

Question

A friend of mine is giving a dinner party. His current wine supply includes 11 bottles of zinfandel, 10 of merlot, and 12 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?
ways

(b) If 6 bottles of wine are to be randomly selected from the 33 for serving, how many ways are there to do this?
ways

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

(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.)

Explanation / Answer

a) If order is important then we will use permutation.

Number of ways to serve 3 bottles of zinfandel = 11P3 = 11! / (11 - 3)! = 990

b) Number of ways to choose 6 bottles of wine = 33C6 = 33! / (6! * (33-6)!) = 1107568

c) Number of ways to obtain two bottles of each variety = 11C2 * 10C2 * 12C2 = 55 * 45 * 66 = 163350

d) P(two bottles of each variety being chosen) = 163350 / 1107568 = 0.147

e) P(all of them are of same variety) = P(all are zinfandel) + P(all are merlot) + P(all are cabernet)

                                              = 11C6/33C6 + 10C6/33C6 + 12C6/33C6

                                              = 462/1107568 + 210/1107568 + 924/1107568

                                              = 0.001

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