How To Solve Cicular Permutation Questions Quickly

How to solve Circular Permutation Questions Quickly

Circular Permutation of an arrangement in a closed loop can be calculated by the formula ,(n-1) ! where n is the number of items.

How to solve Circular Permutation Questions Quickly

Definition & How to Solve Circular Permutation

  • Combination is an arrangement of objects where order does not matter.
  • There are also arrangements in closed loops, called circular arrangements

Type 1: Find the greatest or smallest number.

Question 1. Anuradha invited her 5 friends for dinner. In how many ways she can make them sit around a circular table?

Options

      1. 120
      2. 12
      3. 24
      4. 72

Solution:    Number of arrangements possible = (n − 1)!

= (5 – 1)!

= 4!

= 24

Correct option: 3

Question 2. A gardener wants to plant some Neem trees around a circular pavement. He has 7 different size of Neem trees. In how many different ways can the Neem tree be planted?

Options

      1. 2520
      2. 2400
      3. 5040
      4. 720

Solution:    Number of arrangements possible = (n − 1)!

= (7 – 1)!

= 6!

= 720

Correct option: 4

Question 3. In how many ways can 4 men and 4 women be seated at a circular table so that no 2 women sits together?

Options

      1. 414
      2. 120
      3. 240
      4. 144

Solution:     4 men may be seated in 3! ways, leaving one seat empty. Then at remaining 4 seats, 4 women can sit in 4! ways.

= 3! * 4!

= 3 * 2 * 1 * 4 * 3 * 2* 1

= 6 * 24

= 144

Correct option: 4

Type 2: When clockwise and anticlockwise

Question 1. How many different garlands can be made using 10 flowers of different colors?

Options

      1. 181440
      2. 362880
      3. 145690
      4. 5040

Solution:    Number of arrangements possible = \frac{1}{2} × (n-1) !

= \frac{1}{2} × (10-1) !

= \frac{1}{2} × 9 !

= \frac{1}{2} × 362880

= 181440

Correct option: 1

Question 2. How many necklace of 10 beads each can be made from 20 beads of different colors?

Options

      1. \frac{10!}{19^2}
      2. \frac{{19!}^2)}{10!}
      3. \frac{19!}{19^2}
      4. \frac{10!}{10^2}

Solution:    In case of necklace the clockwise or anticlockwise arrangements are not different. Therefore, the required ways

= \frac{^{20}P_{10}}{10 ×2}

= \frac{20! }{10! × 10 ×2}

= \frac{{19!}^2)}{10!}

Correct Option : 2

Question 3. In how many ways can 7 different colors beads be threaded in a string?

Options

      1. 3600
      2. 450
      3. 360
      4. 540

Solution:    As necklace can be turned over, clockwise and anti-clockwise arrangements are the same. Therefore, Number of arrangements possible

= \frac{1}{2} × (n-1)!

= \frac{1}{2} × (7-1)!

= \frac{1}{2} × 6!

= 360

Correct option: 3

Read Also – Formula to solve Circular Permutations questions