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28 May 2022, 06:38
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A
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Difficulty:
65%
(hard)
Question Stats:
47% (01:26) correct
53%
(01:51)
wrong
based on 259
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Three dogs and three cats are to be arranged in a circle so that no two dogs are next to each other. If two arrangements are considered different only when the positions of the animals are different relative to each other, how many arrangements are possible?
(A) 12
(B) 18
(C) 24
(D) 36
(E) 60
(A) 12
(B) 18
(C) 24
(D) 36
(E) 60
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29 May 2022, 04:56
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AlevtinaN
AlevtinaN
If the situation were to line up three cats and three dogs such that no two dogs are next to each other, then we would have a first place in line, second, third, fourth, fifth, and sixth. If we took the animal from the back of the line and moved it to the front, all the other animals would move back one spot, and we would have a new arrangement. All we really did was move the animals by one spot in a circle, but it created a new arrangement.
With circle setups, we don't have a first place at the table, second, third, etc; all the seats are equal. So if we move the animals one space to the left, we can't count that as a new arrangement the same way we did with the line. The way to resolve that is to simply place any one element at an arbitrary seat and solve from there.
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28 May 2022, 10:12
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BrentGMATPrepNow
If no two dogs are next to each other, we always need a cat between any two dogs. So the only way to seat these animals is DCDCDC.
On circle questions, it's easiest to start by simply choosing a spot for one of the elements.
Let's put a cat at seat 1.
Seat 2 must be a dog. How many options do we have for seat 2? There are three dogs and we can choose any of them, so three options.
Seat 3 must be a cat. How many options do we have for seat 3? We already have one cat seated, so two options.
Seat 4 must be a dog. How many options do we have for seat 4? We already have one dog seated, so two options.
Seat 5 must be a cat. How many options do we have for seat 5? We already have two cats seated, so one options.
Seat 6 must be a dog. How many options do we have for seat 6? We already have two dogs seated, so one options.
3*2*2*1*1 = 12
Answer choice A
