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Geometrical Permutations - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • CIRCULAR PERMUTATIONS, DIFFERENT CASES OF GEOMETRICAL ARRANGEMENTS is considered one of the most asked concept.

  • 17 Questions around this concept.

Solve by difficulty

The number of ways, in which 5 girls and 7 boys can be seated at a round table so that no two girls sit together, is:

A group of 10 people, including 3 managers and 7 employees, is going to sit at a round table for a meeting. If the managers must sit together, in how many different ways can they be arranged?

 

A group of 8 friends, including 4 men and 4 women, is going to sit at a rectangular table. If the men and women must alternate seats, in how many different ways can they be seated?

A group of 12 friends, including 5 men and 7 women, is going to sit at a rectangular table. If the men and women must alternate seats, and the two men and two women must sit at the corners, in how many different ways can they be seated?

 

Concepts Covered - 2

CIRCULAR PERMUTATIONS

Let’s say there is a round table with 6 chairs all identical and 6 persons have to sit. For the first person there is only one choice to make as all chairs are identical, so wherever he may sit doesn’t matter. Now when 1st person has sit, 2nd person with respect to 1st have five choices to sit, directly opposite or left or 2nd from left or right or 2nd from right to 1st person, in the same ways 3rd person will have 4 choices, 4th person will have 3 choice, 5th person will have 2 choices, and last person 1 choice, so in that way total 5 x 4 x 3 x 2 x 1 = ( 6 - 1 )! permutations are possible. 

We can generalize this result as an object that can be arranged along with a circular table in (n-1)! Ways

 

Example: In how many ways 7 people can be arranged along a circular table having 7 identical chairs.

Solution: using the above concept, it can be done in (7-1)! =  6!.

 

Necklace, Garland Type Questions

If clockwise and anticlockwise permutations are same in a circular permutation, (as in case of garlands or necklace formation), the number of permutations becomes (½)(n -1)!

Since as in the case of necklace and garland if we flip the bead or garland then anticlockwise arrangements become clockwise but they are identical because the objects are identical, hence two arrangements are reduced to one causing the total number of permutations to be halved.

Example: Find the ways in which 10 different beads can be arranged to form a necklace?

Solution: Using circular permutations the total number of permutations = (10-1)! 

Now since clockwise and anticlockwise arrangements give the same permutation so the total number of permutations becomes (½)(9!)

DIFFERENT CASES OF GEOMETRICAL ARRANGEMENTS

If clockwise and anticlockwise permutations are same in a circular permutation, (as in case of garlands or necklace formation), the number of permutations becomes (½)(n-1)!.

Since as in the case of necklace and garland if we flip the bead or garland then anticlockwise arrangements become clockwise but they are identical because the objects are identical, hence two arrangements are reduced to one causing the total number of permutations to be halved.

Example: Find the ways in which 10 different beads can be arranged to form a necklace?

Solution: Using circular permutations the total number of permutations = (10-1)! 

Now since clockwise and anticlockwise arrangements give the same permutation so the total number of permutations becomes (½)(9!)

Study it with Videos

CIRCULAR PERMUTATIONS
DIFFERENT CASES OF GEOMETRICAL ARRANGEMENTS

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