Circular Permutations

- are arrangements of elements in a circular pattern.

Formula:

Example 1:

• In how many ways can 5 people be seated around a circular table?

n=5

Formula: (n-1)!

(5-1)!

4!

4.3.2.1

= 24

• In how many ways can 8 girls be seated in a dinner table?

n=8

Formula: (n-1)!

(8-1)!

7!

= 5,040

* There are two cases of circular permutation;*

(a) If clockwise and anti clock-wise orders are different, then total number of circular-permutations is given by (n-1)!

(b) If clock-wise and anti-clock-wise orders are taken as not different, then total number of circular-permutations is given by (n-1)!/2!

Example 2:

How many necklace of 12 beads each can be made from 18 beads of different colours?

clock-wise and anti-clockwise arrangement s are same.

total number of circular–permutations:

¹⁸P₁₂/2x12

=18!/(6 x 24)

= 4.45x10 to the power of 13

* Some permutation problem have restrictions that affect the

answer and the process of solving it

• In how many ways can 8 people be seated around a circular table if Edgar and EJ refuse to sit next to one another?

STEP 1: Calculate total arrangements that is given in the problem.

n=8

Formula: (n-1)!

(8-1)!

7!

=5040

STEP 2: Calculate the number without following the restrictions, for this calculate the number when Edgar and Ej sit together.

*Edgar and Ej is going to be a one group because they sit together, so it makes (2!)

*added to the people left. (6!)

= 7!

(n-1)

(7-1)! =6!・2! = 1440

STEP 3: Calculate the number following the restrictions, for this calculate the number when Edgar and Ej is not sitting together.

* Subtract the total number of arrangement and the total number when Edgar and Ej is sitting together .*

5040-1440 = 3600

Permutations with Case Restrictions

Example 1:

*In how many 5 letter “words” are possible using the letters in BUFFALO?

*used the dash method

*count how many letters are their in the word BUFFALO. so theres 7 words.

*find letters that are repeated and divide the total number of with the number of letters that are repeated. and theres 2 repeated letters.

(7・6・5・4・3) ÷ 2 = 1,260

• Theres a set of basket of fruits being arranged in a cabinet. The first set has 3 oranges, the second has 4 bananas and the third set has 5 apples. In how many ways can the basket of fruits can be arranged if they should be kept in a same cabinet together?

○ - oranges

▲- apple

～ - bananas

～～～～ 4!

▲▲▲▲▲ 5!

○○○ 3!

*multiply all the number of quantity of the given number of fruits and the number of item that are given.

4! ・ 5! ・3! ・3! = 103,680