# Tips and Tricks and Shortcuts for Circular Permutation Questions

## 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.

### Type 1: When clockwise and anticlockwise arrangements are different.

• 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. 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. 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. 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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