Problem Solving

Ques: 3 marbles are placed at each corner of a traingle. 2 arrangements of the marbles are considered different only if the relative position of the marbles are different. How many different ways are there to arrange the 3 marbles?

1. 1
2. 2
3. 3
4. 4
5. 6

relative position is considered with the permuted (ordered) sets
3P2=3!=6
edit: 3P2/3=2

bharti.2010 wrote:Ques: 3 marbles are placed at each corner of a traingle. 2 arrangements of the marbles are considered different only if the relative position of the marbles are different. How many different ways are there to arrange the 3 marbles?

1. 1
2. 2
3. 3
4. 4
5. 6

IMO B

This is similar to arranging 3 marbles in a circle (the circle would be the circle that circumscribes the triangle).

Also, 2 arrangements are different only if the relative positions of the marble are different. So one marble's position is fixed. Total number of arrangements = number of arrangements of the remaining two marbles [spoiler]= 2! = 2. Or the formula for computing the number of ways of arranging n object in a circle = (n-1)! i.e. (3-1)! = 2 [/spoiler].

If we consider A, B, C as the 3 marbles, the only two unique arrangements are ABC and ACB. Each of the other 4 arrangements can be obtained by rotating one of these two triangles.

Yes Santosh you are correct. Answer is 2.
I got confused in the statement which says " relative position of the marbles are different".
Thanks for your explanatio.

There are only 2 ways as one marble top be treated fixed. The other two can be arranged in 2!Q ways = 2 ways.
(B) is answer.

yes Santosh, i missed that it's circular permutation
therefore, 3P2/3=2
choice b

santhoshsram wrote:IMO B

This is similar to arranging 3 marbles in a circle (the circle would be the circle that circumscribes the triangle).

Also, 2 arrangements are different only if the relative positions of the marble are different. So one marble's position is fixed. Total number of arrangements = number of arrangements of the remaining two marbles [spoiler]= 2! = 2. Or the formula for computing the number of ways of arranging n object in a circle = (n-1)! i.e. (3-1)! = 2 [/spoiler].

If we consider A, B, C as the 3 marbles, the only two unique arrangements are ABC and ACB. Each of the other 4 arrangements can be obtained by rotating one of these two triangles.

yes got it right this time. (3-1)! -- relative positions is the key