How To Solve Cicular Permutation Questions Quickly

Type 2: When clockwise and anticlockwise

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

Options

1. 1. 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. 1. 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!}{10!}

                     Now the beads can be arranged in the Circular Fashion in (20-1) = 19 ! ways

                     Required number of ways = \frac{19!}{10!} \times 19!

= \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. 1. 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