(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 29 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?

(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) There are 6 ways to serve 3 bottles of zinfandel if order is important. b) There are 594,914 ways to randomly select 6 bottles from 29. c) There are 9 ways to obtain 2 bottles of each variety when 6 bottles are randomly selected. d) The probability that all 6 selected bottles are the same variety is approximately 0.000005.

Explanation:

a) If order is important, there are 3! = 3 × 2 × 1 = 6 ways to serve 3 bottles of zinfandel.

b) When randomly selecting 6 bottles from 29, the number of ways is given by the combination formula: C(29, 6) = 29! / (6! × (29-6)!) = 594,914 ways.

c) If 6 bottles are randomly selected, and we want to obtain 2 bottles of each variety, we have to consider the number of ways to choose 2 bottles from each variety and then multiply the results. There are C(3, 2) ways to choose 2 bottles from each variety, so the total number of ways is C(3, 2) × C(3, 2) = 3 × 3 = 9 ways.

d) The probability that all 6 randomly selected bottles are the same variety is the probability of selecting 6 bottles of the same variety divided by the total number of ways of selecting 6 bottles. Since there are 3 varieties and each variety has C(3, 6) = 1 ways to choose 6 bottles, the probability is 3/594,914 ≈ 0.000005.

Learn more about combinations and probability here:

https://brainly.com/question/33890198

#SPJ11