A friend of mine is giving a dinner party. His current wine supply includes 11 b
ID: 3369279 • Letter: A
Question
A friend of mine is giving a dinner party. His current wine supply includes 11 bottles of zinfandel, 12 of merlot, and 9 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 32 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
Solution:
number of zinfandel bottles = 11
number of merlot bottles = 9
number of cabernet bottles = 8
( a )
number of zinfandel bottles = 11
number of bottles to be served = 3
serving order is important .so, serving 3 bottles out of 11 zinfandel bottles with serving order into consideration comes under permutation
number of ways of serving 3 zinfandel bottles when order is important = 11P3 = 11! / (11?3)! = 990
( b )
number of ways of selecting 6 bottles out of 32 bottles = 32C6 = 32! / (6!(32?6)!) = 906192
( c )
number of ways of selecting 2 zinfandel bottles out of 11 of them = 11C2 = 11! / (2!(11?2)!) =55
number of ways of selecting 2 merlot bottles out of 12 of them = 12C2 = 12! / (2!(12?2)!) =66
number of ways of selecting 2 cabernet bottles out of 9 of them = 9C2 = 9! / (2!(9?2)!) =36
number of ways of selecting two of each variety = 11C2 * 9C2 * 8C2 = 55 * 66 * 36 = 130680
( d )
number of ways of selecting 6 bottles out of 32 bottles = 906192
number of ways of selecting 2 bottles of each variety = 130680
probability of selecting two of each variety = 130680 / 906192 = 0.1442
( e )
probability of selecting 6 bottles of zinfandel = 11C6 / 32C6 = 462 / 906192= 0.0005
probability of selecting 6 bottles of merlot = 12C6 / 32C6 = 924/ 906192= 0.0010
probability of selecting 6 bottles of cabernet = 9C6 / 32C6 = 84 / 906192= 0.00009
probability of selecting 6 bottles of same kind = 0.0005+0.0010+0.00009= 0.00159
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