A friend of mine is giving a dinner party. His current wine supply includes 12 b

Question

A friend of mine is giving a dinner party. His current wine supply includes 12 bottles of zinfandel, 8 of merlot, and 10 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 30 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.)

Explanation / Answer

a)number of ways =N(selectinng 3 out of 12 and then arranging) =12P3 =12!/9! =12*11*10=1320

b) number of ways =30C6 =30!/(6!*(30-6)!) =593775

c) number of ways =N(Selecting 2out of 12 bottles of zinfandel and selecting 2 out of 8 of merlot and selecting 2 out of 10 of cabernet) =12C2 *8C2 *10C2 =83160

d) probability =83160/593775=0.140

e) number of ways all are same =N(all 6 are zinfandel +all 6 are merlot+all 6 are cabernet)

=12C26 +8C6 +10C6 =139

hence probability=139/593775 =0.0002

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