A man wants to visit at least two of the four cities A, B, C

Question

A man wants to visit at least two of the four cities A, B, C and D. How many travel itineraries can he make? All cities are connected to one another.

(A) 24
(B) 6
(C) 60
(D) 12
(E) None of the above

Bunuel · Accepted Answer

The # of ways to visit 2 cities when order matters = \(P^2_4=\frac{4!}{(4-2)!}=12\): AB, BA, AC, CA, AD, DA, BC, CB, BD, DB. CD, DC.

The # of ways to visit 3 cities when order matters = \(P^3_4=\frac{4!}{(4-3)!}=24\): ABC, ACB, BAC, BCA, CAB, CBA, ...

The # of ways to visit 4 cities when order matters = \(P^4_4=\frac{4!}{(4-4)!}=24\): ABCD, ABDC, ADCD, ...

12 + 24 + 24 = 60.

Answer: C.

P.S. Please read carefully and follow: https://gmatclub.com/forum/rules-for-po ... 33935.html Pay attention to rules 2, 3, 5, 7 and 10. Thank you.

JusTLucK04 · Answer

We can also say:

4C2*2! + 4C3*3! + 4C4*4!

I am quite confused with this P n C terminologies..on where to use...so I go intutively..i.e First select the elements and then shuffle them with factorial (If required)...Veritas has an excellent guide on combinatorics..which combined with Bunuels Q bank is enough for 700+ Q on this topic

Bunuel · Answer

This is correct.  C^2_4*2!  is the same as  P^2_4 .

SonofAnarchy · Answer

Why not 4C2x4C3x4C4?

mathivanan · Answer

At least 2 means 2 or more
2 cities can be selected from 4 in 4C2 ways and these two cities, say, A and B have two distinct itineraries AB and BA
That can be done in 4C2*2! or 4P2 ways=12
Similarly, 3 cities in 4C3*3! or 4P3 ways=24; 4 cities in 4C4*4! or 4P4 ways=24
Total 12+24+24=60 ways.
The correct option is C

Ashishsteag · Answer

Travel itinerary means you can start your travel from any city-say you start from B and then moved to A and other way round you can start from A and move to B-so you have two different itineraries.Itineraries literally means planned routes and so you have two different routes as I explained.

chelawatrishabh · Answer

In this question  ALL THE CITIES ARE CONNECTED TO ONE ANOTHER , .How you consider this piece of information.I am not getting this part. I am supposed to select at least two cities. could you please just give me a brief idea. how to solve this type of questions.

BillyZ · Answer

Bunuel  Why we need 4C2* 2! + 4C3* 3!  + 4C4* 4!  as highlighted?

hindu1988 · Answer

hazelnut 
Ways to choose cities x ways to travel 
At least 2 cities —> 6 ways (2 cities chose out 4) x 2! (Ways to travel to cities) = 12
3 cities —> 4 x 3! =24
4 cities —> 1 x 4! = 24 
Total = 60 (C)

saby1410 · Answer

VeritasKarishma  
why we need arrangement in this question. doesn't this line(All cities are connected to one another) means a-b-c-d
Did i missed one thing that it's not mentioned from where to start so we can go for AB as well as BA, same for others 
For atleast 2 cities

KarishmaB · Answer

"All cities are connected to one another."
means each city is connected to all other cities. 
So A is connected to B, C and D independently. So he can start from any city and then go to any other from there.

It does not mean that A is connected to only B which in turn is connected to only C which in turn is connected to only D.

Adit_ · Answer

I couldn't really understand what is meant by travel itinerary. I thought the question meant no.of ways to travel to atleast 2 location, how are we assuming the order matters here? Kindly help me out Bunuel

Bunuel · Answer

“Travel itinerary” means a planned sequence of stops, so the order you visit cities is part of the itinerary. Visiting A then B is a different itinerary from visiting B then A because the day by day plan is different, even though the set of cities visited is the same. The line “all cities are connected” just tells you any order is feasible, so you are allowed to count all ordered sequences.

Adit_ · Answer

ahh yes gotcha. So can I keep it a rule that any questions specifically asking for "arrangements", "itineraries" of sort will always encompass permutations?

Bunuel · Answer

Usually, yes: itinerary, schedule, or ranking implies order matters, so use permutations.

In general, “permutation” and “arrangement” are used interchangeably: order matters.

In general, “combination,” “group,” and “selection” are used interchangeably: order does not matter.

But it’s still context dependent, so always check whether order changes the outcome.

A man wants to visit at least two of the four cities A, B, C

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A man wants to visit at least two of the four cities A, B, C

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