A friend of mine is giving a dinner party. His current wine supply includes 9 bo

Question

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

Explanation / Answer

9 bottles of zinfadel

11 bottles of merlot

8 bottles of carbernet

All are different wineries.

a) If he wants to serve 3 bottles of zinfandel, and total 9 bottles of zinfandel

So total number of ways are 9P3 = 9 * 8 * 7 = 503

b) If 6 bottles of wine are to be randomly selected from 28 for serving

If order is not important then number of way are 28C6 = 376740

If order is important then number of ways are 28P6 = 28*27*26*25*24*23 = 271252800

c) If 6 bottles are randomly selected and two bottles of each variety.

Number of ways to selecting 2 bottles from zinfandel = 9C2 = 36

Number of ways to selecting 2 bottls from merlot = 11C2 = 55

Number of ways to selecting 2 bottles from cabernet = 8C2 = 28

So total number of ways to selecting two bottles from each variety = 36 * 55 * 28 = 55440

d) Total bottles are 9 + 11 + 8 = 28

If 6 bottles are selected from 28 bottles so number of ways are 28C6 = 376740

total number of ways to selecting two bottles from each variety = 55440

So probability that this result in two bottles of each variety being chosen = 55440 / 376740 = 0.147

e) If 6 bottles are randomly selected and we have to find probability that all of them are the same variety.

If 6 bottles are selected from 28 bottles so number of ways are 28C6 = 376740

Number of ways to selecting 2 bottles from zinfandel = 9C2 = 36

Number of ways to selecting 2 bottls from merlot = 11C2 = 55

Number of ways to selecting 2 bottles from cabernet = 8C2 = 28

So number of ways they are same variety = 36 + 55 + 28 = 119

So the probability that all of them are same variety = 119 / 376740 = 0.0003

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