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, 10 of merlot, and 12 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? x ways (b) If 6 bottles of wine are to be randomly selected from the 31 for serving, how many ways are there to do this? x 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

Please note for combinations nCr = n! / [(n-r)!*r!]

(a) Since order is important, it means arrangements or permutations are needed, so 3 bottles can be chosen from 6 and arranged, which is 6C3 * 3! = {6! / [(6-3)!*3!] }* 3! = 20 * 6 = 120 ways

____________________________________________________________________________

(b) Choosing 6 bottles from 31 = 31C6 = 31! / [(31-6)!*6!] = 31!/(25!*6!)

= (31 * 30 * 29 * 28 * 27 * 26 * 25!)/(25! * 6 * 5 * 4 * 3 * 2 * 1) = 736,281 ways

______________________________________________________________________________

(c) We need to choose 2 from 9 of zinfandel, 2 from 10 of merlot and 2 from 12 of cabernet. Therefore total number of ways = 9C2 * 10C2 * 12C2 = 36 * 45 * 66 = 106,920 ways.

_________________________________________________________________________________

(d) Probability = Favourable outcomes/Total outcomes

From (d) The favourable outcomes of choosing 2 bottles of each type = 106920 and from (c) the total outcomes of choosing 6 bottles out of 31 = 736281.

Therefore the required probability = 106920/736281 = 0.145

__________________________________________________________________________________

(e) Probability = Favourable outcomes/Total outcomes

Favourable outcomes = 6 bottles chosen are of the same variety = 6 bottles of zinfardel or 6 bottles of merlot or 6 bottles of cabernet (or means these are different events) = 9C6 + 10C6 + 12C6 = 84 + 210 + 924 = 1218

Total outcomes = choosing 6 bottles out of 31 = 31C6 = 736281

Therefore the required probability = 1218/736281 = 0.002

__________________________________________________________________________________

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.