Tips and Tricks and Shortcuts for Circular Permutation Questions

December 2, 2022

Tips and Tricks for Circular Permutations Questions

The Calculation of the number of ways to arranging the objects, items or peoples in closed loop or in a circular manner referred as Circular Permutation. In this Page Tips and Tricks for Circular Permutation is given.

Tips and Tricks for Circular Permutation

Circular Permutation are arrangements in the closed loops.

If clockwise and anti-clock-wise orders are different, then total number of circular-permutations is given by (n-1)!
If clock-wise and anti-clock-wise orders are taken as not different, then total number of circular-permutations is given by $\frac{(n-1)!}{2!}$
While Solving Typical Problems divide Problems in Small Parts then think upon it.

Trick: Number of circular permutations (arrangements) of n distinct things when arrangments are different = (n−1)!

Question 1. In how many ways can 5 girls be seated in a circular order?

Options:

1. 1. 45
  2. 24
  3. 12
  4. 120

Solution Number of arrangements possible = (5 − 1)!

= 4!

= 4 x 3 x 2 x 1

= 24

Correct option: 2

Question 2 . Determine the number of ways in which 5 married couples are seated on a Circular Round table if the spouses sit opposite to one another.

Options

1. 1. 120
  2. 320
  3. 384
  4. 387

Solution 5 Married couples means we have to arrange 10 peoples

First women can be placed anywhere in a circular Round Table and her husband in 1 way.

Second women can be placed in 8 ways and her husband in 1 way.

Third woman can be placed in 6 ways and her husband in 1 way.

Fourth women can be placed in 4 ways and her husband in 1 way.

Fifth women can be placed in 2 ways and her husband in 1 way.

Total number of ways $1\times 8\times 1 \times 6\times 1 \times 4\times 1 \times 2 \times 1 = 384$ Ways

Correct option: 3

Type 2: When clockwise and anticlockwise arrangements are not different

Tips & Trick: Number of circular permutations (arrangements) of n distinct things when arrangements are not different = $\frac{1}{2}$ × (n−1)!

Question 1. In how many ways can 8 beads can be arranged to form a necklace?

Options

1. 1. 2520
  2. 5040
  3. 360
  4. 1200

Solution Since In formation of Necklace the Clockwise and Anti-clockwise Arrangements are Same So we divide by 2

Number of arrangements possible = $\frac{1}{2}$ × (n−1)!

$\frac{1}{2}$ × (8−1)!

$\frac{1}{2}$ × 7!

$\frac{1}{2}$ × 5040 = 2520

Correct option: 1

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