Assume there are 10 cities that are finalists for a “best place for millennials
ID: 2922929 • Letter: A
Question
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., 10th best, 9th best, ..., 2nd best, best city. The cities are:
Akron, Albany, Albuquerque, Annapolis, Atlanta, Austin, Dallas, Portland, Sacramento, and Toledo
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 9th 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?
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
Answer to part(a)
The no. of ways top 6 spots can be filled by cities with only "A" cities is as follows.
First city can occupy any of the 1st 6 spots, so 6 ways for placing 1st city.
Second city can occupy any of the remaining 5spots, so 5 ways for placing 2nd city.
Similarly, third city can occupy any of the remaining 4 spots, so 4 ways for placing 3rd city.
....
So,no.of ways top 6 spots can be occupied 6*5*4*3*2*1=6!=720
Similarly, last 2 spots can be filled in 2! ways.
And for fillin 7th and 8th positions, again it can be done in 2! ways.
So, total no. of ways in which top 6 are occupied by cities with name starting from "A") and last 2 with Sacramento and Toledo is 6! *2! * 2!= 720*2*2=2880
answer to part b
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?
simply, total no. of ways calculated in part(a) / no. of ways 10 cities can occupy 10 places
=6!*2!*2! / 10! ( explained in part a)
=4/7*8*9*10
=1/1260 =.0793%