Math Problem Statement
A friend of mine is giving a dinner party. His current wine supply includes 11 bottles of zinfandel, 13 of merlot, and 7 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? 990
Correct: Your answer is correct. (b) If 6 bottles of wine are to be randomly selected from the 31 for serving, how many ways are there to do this? 1653848
Incorrect: Your answer is incorrect. (c) If 6 bottles are randomly selected, how many ways are there to obtain two bottles of each variety? 91080
Incorrect: Your answer is incorrect. (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.) 0.153
Incorrect: Your answer is incorrect. (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.)
Solution
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Math Problem Analysis
Mathematical Concepts
Combinatorics
Probability
Permutations
Combinations
Formulas
Combinations formula: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)
Permutations formula: \(P(n, k) = \frac{n!}{(n - k)!}\)
Probability formula: \(P = \frac{\text{Favorable outcomes}}{\text{Total outcomes}}\)
Theorems
Basic principles of probability and combinations
Suitable Grade Level
Grades 11-12 or College Level
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