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.

Uva@90 wrote:A,B,C,D,E go to a restaurant. there are five chairs in a Circular arrangement. in how many ways they can sit in the circular arrangement such that A and B don't want to sit together ?

A) 10
B) 12
C) 16
D) 20
E) 24

Once A has been placed at the table, we get:
Number of options for the seat to the left of A = 3. (Of the 4 remaining people, anyone but B.)
Number of options for the seat to the right of A = 2. (Of the 3 remaining people, anyone but B.)
Number of ways to arrange the 2 remaining people = 2! = 2.
To combine these options, we multiply:
3*2*2 = 12.

The correct answer is B.

You can really just think of this as line with limitations for each spot:



A -> Fix the spot
Right of A -> 3 options that are not B
Left of A -> 2 remaining options that are not B
Spot 4 -> 2 options --- B and last one not chosen
Last -> 1 -- the last remaining letter


1 x 3 x 2 x 2 = 12

Uva@90 wrote:A,B,C,D,E go to a restaurant. there are five chairs in a Circular arrangement. in how many ways they can sit in the circular arrangement such that A and B don't want to sit together ?

A) 10
B) 12
C) 16
D) 20
E) 24

OA B

Circular arrangement is (n-1)! after that how should i restrict A and B ?

Regards,
Uva.

Total arrangements = A and B sit together + A and B don't sit together

Total arrangements = 4! = 24

Lets find arrangement of A and B sitting together
We have A B C D E
Lets group A and B into one and call it G

We now have G C D E
number of ways to arrange these in a circle = 3 x 2 x 1 = 6
Since G can have 2 outcome AB or BA we need to mulitple by 2
therefore A and B sit together = 6 x 2 = 12

Total arrangements = A and B sit together + A and B don't sit together
24 = 12 + A and B don't sit together
A and B don't sit together = 12
ans = b

theCEO wrote:

Uva@90 wrote:A,B,C,D,E go to a restaurant. there are five chairs in a Circular arrangement. in how many ways they can sit in the circular arrangement such that A and B don't want to sit together ?

A) 10
B) 12
C) 16
D) 20
E) 24

OA B

Circular arrangement is (n-1)! after that how should i restrict A and B ?

Regards,
Uva.

Total arrangements = A and B sit together + A and B don't sit together

Total arrangements = 4! = 24

Lets find arrangement of A and B sitting together
We have A B C D E
Lets group A and B into one and call it G

We now have G C D E
number of ways to arrange these in a circle = 3 x 2 x 1 = 6
Since G can have 2 outcome AB or BA we need to mulitple by 2
therefore A and B sit together = 6 x 2 = 12

Total arrangements = A and B sit together + A and B don't sit together
24 = 12 + A and B don't sit together
A and B don't sit together = 12
ans = b

Thanks CEO,

This is the approach I was looking for.

Regards,
Uva.

like CEO's approach. it is simple & crystal clear.

we use the formula as (n-1)! here

(5-1)! - 2 * (4-1)!
=4! - 2 * 3!
=24 - 2 * 6
=24-12
=12