Question 4 (10 puts) Assume there are 10 cities that are finalists for a \"best
ID: 2922849 • Letter: Q
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Question 4 (10 puts) Assume there are 10 cities that are finalists for a "best place for millennials to live" list in a popular magazine. The cities' merits are nearly equivalent, so all the names are placed in a hat and drawn one by one to find e.g., 10* best, 9* best, ., 2ad best, best city. The cities are Akron, Albany, Albuquerque, Annapolis, Atlanta, Austin, Dallas, Portland, Sacramento, and Toledo a) It's important to the editor that the top 6 spots not be filled with only cities whose name starts with the letter "A", and that neither the 9&h; or 10th place cities be filled with Sacramento and Toledo. How many ways could it happen that the top 6 spots be filled with only "A" cities while the last 2 spots be filled with Sacramento or Toledo? (5 pnts) b) What is the likelihood of the top six spots being filled with only "A" cities and the last 2 spots being filled with Sacramento and Toledo, using the hat method of picking winners? (5 pnts)Explanation / Answer
A) 6 Cities start with Letter A. The other four are Dallas, Portland, Sacramento, Toledo.
We want to find the no. ways Top 6 are cities which start with letter 'A' and 9th and 10th place being accompanied by Sacramento or Toledo.
1)The 6 cities staring with A can take the first 6 spot in 6C6 * 6! i.e. selecting the 6 cities with A in top 6 position(6C6) and arranging them in 6 different places (6!) = 720 ways
2)7th and 8th have to be Portland and Dallas. Since Top 6 are taken Dallas and Portland can take 7th and 8th in 6C6*2! = 2 ways (i.e. arranging them as Portland,Dallas or as Dallas,Portland)
3)Sacramento or Toledo can take the final 2 spots in 2C2* 2! = 2 ways.
No. of ways in which all 3 happen together is 720*2*2 = 2880 ways
B) The probability of the above arrangement happening would be (No. ways of in which Top 6 is cities with A and Sacramento or Toledo at last two positions) / Total no. of diferent arrangements of 10 cities
Total ways in which we can arrange 10 cities among themselves is 10! = 3628800
Likelihood of top six filled by cities starting with 'A' and Sacramento or Toledo at last two positions by the hat method is 2880 / 3628800 = 0.000793650
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