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# Jack is making a list of his 5 favorite cities. He will choose 3 citie

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Jack is making a list of his 5 favorite cities. He will choose 3 citie [#permalink]
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The selection process in the United states is 5C3 and that in Europe is 3C2. (Since its a selection process we use combinations).

Therefore we get [(5*4)/(2*1) * 3] = 30 ---- Upon simplification. (1)

The Trap here would be to consider the above as the answer.

At the end of the statement it says that "different lists of cities, ranked from first to fifth".

Therefore the Cities can be arranged in 5! Ways which is 5*4*3*2*1 = 120 (2)

From (1) and (2) we get 30*120 = 3600.

Therefore option D is the answer.

Originally posted by Kritesh on 09 May 2017, 13:17.
Last edited by Kritesh on 21 May 2017, 03:33, edited 1 time in total.
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Re: Jack is making a list of his 5 favorite cities. He will choose 3 citie [#permalink]
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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

If the ranking doesn’t matter, there are 5C3 x 3C2 lists of 5 cities. However, since, for each list of the 5 cities, there are 5! ways to rank the cities from first to fifth, the total number of ranked lists is:

5C3 x 3C2 x 5!

(5 x 4 x 3)/(3 x 2) x 3 x 120

10 x 3 x 120 = 3,600

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Re: Jack is making a list of his 5 favorite cities. He will choose 3 citie [#permalink]
(1st)Needs to Choose which 3 American Cities and which 2 European Cities will be ranked.

Out of the 5 possible American cities, he can choose Groups of 3 in:

"5 choose 3" = Number of Ways

AND

For Each Combination of 3 Unique American Cities chosen, he then picks 2 out of 3 European Cities. He does this in:

"3 choose 2" = Number of Ways

All the Different Combinations of 3 Unique American Cities and 2 Unique European Cities that can be chosen is found by Multiplying:

(5! / 2!3!) * (3! / 2!1!) = 30 Different Ways to Choose

AND

(2ND) For Each Unique Combination of 5 Cities chosen, he must Arrange the Cities from Least - to - Worst. If there is N Unique Elements, they can be Arranged in N! Ways.

5! = 120

(30) * (120) = 3,600

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Re: Jack is making a list of his 5 favorite cities. He will choose 3 citie [#permalink]
there are 3 parts to the question:
1. ways to choose 3 cities from 5 United states : 5c3 ;10
2. 2 cities from list of 3 Europe ; 3c2; 3
3. Arrangement list of 5 cities in different ways ; 5! ; 120
total ways ; 10*3*120 ; 3600
option D