Problem Solving

At a meeting of the 7 Joint Chiefs of Staff, the Chief of Naval Operations does not want to sit next to the Chief of the National Guard Bureau. How many ways can the 7 Chiefs of Staff be seated around a circular table?
A-120
B-480
C-960
D-2520
E-5040
OA-B
These type of circular arrangement prob trouble me a lot.Experts help me how to tackle this and build a workable concepts about this.pl help.

To count CIRCULAR arrangements:

1. Place one element in the circle.
2. Count the number of ways to arrangement the REMAINING elements RELATIVE to the first element.

sandipgumtya wrote:At a meeting of the 7 Joint Chiefs of Staff, the Chief of Naval Operations does not want to sit next to the Chief of the National Guard Bureau. How many ways can the 7 Chiefs of Staff be seated around a circular table?
A-120
B-480
C-960
D-2520
E-5040

Once the Chief of Naval has been placed at the table, we get:
Number of options for the seat to the left of the Chief of Naval = 5. (Of the 6 remaining people, anyone but the Chief of the National Guard.)
Number of options for the seat to the right of the Chief of Naval = 4. (Of the 5 remaining people, anyone but the Chief of the National Guard.)
Number of ways to arrange the 4 remaining people = 4! = 24.
To combine these options, we multiply:
5*4*24 = 480.

The correct answer is B.

Similar problem:
https://www.beatthegmat.com/circular-arr ... 84925.html

These problems are very easy on the GMAT if you've seen them before: given n people, there (n - 1)! ways to arrange them around a circle. (I'm assuming, of course, that there are no other restrictions.)

I wouldn't expect a basic circular permutation on test day: such a problem is challenging if you've never seen it before, but trivially easy if you have. The test writers don't like those kinds of problems, as you can imagine!

thanks sir.but are these type very common on GMAT?How to build basic knowledge about this?any resource.

Forget conventional ways of solving math questions. In PS, IVY approach is the easiest and quickest way to find the answer.


for n number people, the number of cases for sitting around in a circular table is (n-1)!. Essentially for this question you need to calculate the 'total number of cases - (number of cases the CNO and CNGB sit right next to each other). Remember that CNO can either sit on the right or the left of CNGB, and consider them as a group.


Then it's (7-1)!-2*(6-1)!=6!-2*5!=(6-2)*5!=4*5!=4*120=480, therefore the answer is B


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hi Mitch,
Is there any easier method to do?I really find this topic confusing.can u suggest something easier to remember and tackle all such type of prob.

sandipgumtya wrote:Is there any easier method to do?I really find this topic confusing.can u suggest something easier to remember and tackle all such type of prob.

Generally the permutations problems on the GMAT require one to go somewhat beyond the basic permutations rules in order to figure out the answer.

So, as is the case when one does this problem, often one can't just apply a simple formula or method that one has memorized and expect to get to the answer. Rather one has to go beyond that to analyzing the logic of the situation and figuring out how to apply a formula or general method in combination with some additional reasoning.

It's possible, Sandip, that the problems you are having with these types of questions stem from your understanding of how they work not being clear enough. If you don't really get how permutations work, then you will be rather challenged to figure out how to set things up to get the answer to a question like this one.

So my suggestion to you is to unconfuse yourself by really figuring out how permutations and combinations work.

Maybe this blog post I wrote will help. https://infinitemindprep.com/permutation ... difficult/

Meanwhile, here is another way to answer this question. Maybe you will find this method easier to understand.

You can start by finding all the possible arrangements of the seven people around a circular table. That's pretty simple; it's just (7 - 1)! = 6! = 720.

We now have the maximum number of arrangements of 7 people around a circular table, and also we have a restriction. Because of that restriction the number of possible arrangements will be fewer than 720.

So already we can eliminate all the answer choices that are greater than 720, and we are down to A and B as possibilities.

Now we can get to the answer by subtracting from 720 all the arrangements where the two chiefs who won't sit together are sitting together.

Let's act as if the two of them sitting together were one unit.

If there were not an issue, the Chief of Naval Operations could sit to the right or the left of the Chief of the National Guard Bureau.

So there are two possible units composed of the two of them, Naval Operations Chief on the right and Naval Operation Chief on the left.

For each of those units there are five other chiefs.

So in a sense we have six units, the two of them, and the five others, and there are two different ways to arrange the two of them, so we have two sets of six units, each of which sets we can arrange around the table.

For each of those two sets there are (6 - 1)! = 5! = 120 different ways to arrange them around the table.

So we have 2 x 120 = 240 arrangements in which the Chief of Naval Operations and the Chief of the National Guard Bureau could sit together, if they were both willing to.

However, they are not both willing to sit next to each other. So we need to subtract those arrangements. 720 - 240 = 480.

So the correct answer is B.

Thanks Marty,
Its clearer now.Thanks.