Carcass wrote:
Jack is making a list of his 5 favorite cities. He will choose 3 cities in the United States from a list of 5 candidates. He will choose 2 cities in Europe from a list of 3 candidates. How many different lists of cities, ranked from first to fifth, can Jack make?
A. 30
B. 360
C. 1,800
D. 3,600
E. 6,720
Take the task of creating a list and break it into
stages.
Stage 1: Select 3 US cities
Since the order in which we SELECT the cities does not matter, we can use combinations.
We can select 3 cities from 5 cities in 5C3 ways (10 ways)
So, we can complete stage 1 in
10 ways
Stage 2: Select 2 European cities
Since the order in which we SELECT the cities does not matter, we can use combinations.
We can select 2 cities from 3 cities in 3C2 ways (3 ways)
So, we can complete stage 2 in
3 ways
Stage 3: Arrange the 5 selected cities
We can arrange n unique objects in n! ways.
So, we can arrange the 5 selected cities in 5! ways (120 ways)
We can complete this stage in
120 ways.
By the Fundamental Counting Principle (FCP), we can complete all 3 stages (and thus create our list) in
(10)(3)(120) ways (= 3600 ways)
Answer: D
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Brent Hanneson - founder of Greenlight Test Prep