Circular Permutations Formulas

Sample Questions of Circular Permutations:

Question 1:

A necklace is made using 6 different colored beads. How many distinct necklaces can be formed by rearranging the beads?

Solution :

For a necklace, the arrangement is considered the same up to rotation.

Total number of ways to arrange 6 different beads in a line is 6! (factorial), but we need to divide by 6 to account for the circular arrangement of the necklace.

Hence, the total number of distinct necklaces = (6-1)! = 5! = 120

Question 2:

In how many ways can 4 people and 3 dogs sit around a circular table with alternating seats for humans and dogs?

Solution:

Since the seats are alternating, we need to treat humans and dogs as distinct entities.

First, we will arrange the 4 people in 4! ways around the table. Then, we will arrange the 3 dogs in 3! ways.

However, we must divide by 2 because the dogs are indistinguishable from each other.

Finally, we get the total number of arrangements = \frac{4!\times3!}{2}=36 ways.

Question 3:

A team of 8 basketball players is standing in a circle for a team photo. In how many ways can the players arrange themselves if the captain and vice-captain must stand next to each other?

Solution:

Consider the captain and vice-captain as a single entity.

Now, we have 7 entities (the pair of captain and vice-captain and the other 6 players) to arrange around a circular table.

Number of ways to arrange 7 entities in a circle is (7-1)! = 6!

However, within the captain-vice-captain entity, there are 2 ways to arrange them. Hence, the total number of arrangements = 2 * 6! = 144 ways

Question 4:

In how many ways can 5 books and 4 pens be placed on a circular shelf such that no two pens are adjacent to each other?

Solution:

Let’s assume each pair of book and pen together as a single entity.

Now, we have 5 entities (5 pairs) to arrange around the circular shelf.

Number of ways to arrange 5 entities in a circle is (5-1)! = 4!

However, within each pair, there are 2 ways to arrange the book and pen.

Hence, the total number of arrangements = 2^5\times4!=2^5\times24=768 ways

Question 5:

Five friends – Alice, Bob, Carol, David, and Eve – are going to sit around a circular table. Alice and Bob want to sit together, but David and Eve cannot sit together. In how many ways can they be seated?

Solution:

Let’s treat Alice and Bob as a single entity (AB) and David and Eve as a single entity (DE)

Now, we have 4 entities (AB, Carol, DE) to arrange around the circular table. The number of ways to arrange 4 entities in a circle is (4-1)! = 3!

However, within the AB entity, there are 2 ways to arrange Alice and Bob, and within the DE entity, there are 2 ways to arrange David and Eve.

Hence, the total number of arrangements is 2^2 * 3! = 2^2 * 6 = 24 ways

2^2\times3!=2^2\times6=24 ways