A friend of mine is giving a dinner party. His current wine supply includes 13 bottles of Zinfandel, 11 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?

(b) If 6 bottles of wine are to be randomly selected from the 31 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?

d. The probability is approximately 0.000090 or 0.009%.The number of ways to select items from a set is calculated using combinations.

(a) To calculate the number of ways to select 3 bottles of zinfandel in a specific order, we use the concept of permutations. There are 13 bottles of zinfandel to choose from, and we need to select 3 bottles. Therefore, the number of ways to do this is calculated as 13P3 = 13! / (13-3)! = 13! / 10! = 13 * 12 * 11 = 1,716 ways.

(b) To calculate the number of ways to randomly select 6 bottles of wine from the total of 71 bottles, we can use combinations. There are 71 bottles in total, and we need to select 6 bottles. Therefore, the number of ways to do this is calculated as 71C6 = 71! / (6! * (71-6)!) = 71! / (6! * 65!) = 71 * 70 * 69 * 68 * 67 * 66 / (6 * 5 * 4 * 3 * 2 * 1) = 1,017,939,600 ways.

(c) To calculate the number of ways to select 2 bottles of each variety when s bottles are randomly selected, we can consider each variety separately.

For zinfandel, there are 13 bottles in total, and we need to select 2 bottles. Therefore, the number of ways to do this is calculated as 13C2 = 13! / (2! * (13-2)!) = 13! / (2! * 11!) = 13 * 12 / (2 * 1) = 78 ways.

For merlot, there are 11 bottles in total, and we need to select 2 bottles. Therefore, the number of ways to do this is calculated as 11C2 = 11! / (2! * (11-2)!) = 11! / (2! * 9!) = 11 * 10 / (2 * 1) = 55 ways.

For cabernet, there are 7 bottles in total, and we need to select 2 bottles. Therefore, the number of ways to do this is calculated as 7C2 = 7! / (2! * (7-2)!) = 7! / (2! * 5!) = 7 * 6 / (2 * 1) = 21 ways.

To obtain two bottles of each variety, we need to multiply the number of ways for each variety: 78 * 55 * 21 = 91,980 ways.

(d) To calculate the probability of randomly selecting 6 bottles and obtaining two bottles of each variety, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes.

The number of favorable outcomes is the number of ways to select 2 bottles of each variety: 91,980 ways.

The total number of possible outcomes is the number of ways to select 6 bottles from a total of 71: 1,017,939,600 ways.

Therefore, the probability is calculated as: favorable outcomes / total outcomes = 91,980 / 1,017,939,600 ≈ 0.000090.

So, the probability is approximately 0.000090 or 0.009%.

Know more about permutations here:

https://brainly.com/question/32683496

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