Anna has to visit at least 2 European cities on her vacation trip. If she can visit only London, Paris, Rome, or Madrid,

[Vincen]

Anna has to visit at least 2 European cities on her vacation trip. If she can visit only London, Paris, Rome, or Madrid, how many different itineraries, defined as the sequence of visited cities, can Anna create?

A. 12
B. 36
C. 48
D. 60
E. 72

Answer: D

Source: Veritas Prep

[Scott@TargetTestPrep]

Anna has to visit at least 2 European cities on her vacation trip. If she can visit only London, Paris, Rome, or Madrid, how many different itineraries, defined as the sequence of visited cities, can Anna create?

A. 12
B. 36
C. 48
D. 60
E. 72

Answer: D

Solution:

To visit at least 2 of the 4 given European cities is to visit 2, 3, or all 4 cities. Since each itinerary must be made up of a different sequence of cities, order is important, and thus we have a permutation problem. The permutation formula is nPr = n! / (n - r)!. Let’s determine the number of ways she can visit 2, 3, or all 4 cities.

The number of ways she can visit exactly 2 cities is 4P2 = 4! / 2! = 4 x 3 = 12.
The number of ways she can visit exactly 3 cities is 4P3 = 4! / 1! = 4 x 3 x 2 = 24.
The number of ways she can visit all 4 cities is 4P4 = 4! / 0! = 4 x 3 x 2 x 1 = 24. (Recall that 0! = 1.)

Thus, the total number of ways she can visit at least 2 of the 4 cities is 12 + 24 + 24 = 60.

Answer: D