Question

In how many ways can we arrange 6 different flowers in a circle? In how many ways we can form a garland using these flowers?

Answer 1

Hint: Here we use a formula for circular arrangement of flowers to find the number of ways to arrange them in a circle. Use the concept that a garland is a circular decorative material made with flowers, so keeping in mind the direction of decorative flowers, find the number of ways to form garland.
* Number of ways to arrange n items in a circular manner is given by \[(n - 1)!\]
* A factorial is expanded by the formula \[n! = n \times (n - 1)! = n \times (n - 1) \times (n - 2)!.... = n \times (n - 1) \times (n - 2)....3 \times 2 \times 1\]

Complete step-by-step answer:
We are given 6 different flowers.
We find the number of ways to arrange 6 different flowers by using the formula \[(n - 1)!\]
Substitute the value of \[n = 6\] in the formula
\[ \Rightarrow \]Number of ways to arrange 6 different flowers \[ = (6 - 1)!\]
\[ \Rightarrow \]Number of ways to arrange 6 different flowers \[ = 5!\]
Since we know factorial is expanded by the formula\[n! = n \times (n - 1)! = n \times (n - 1) \times (n - 2)!.... = n \times (n - 1) \times (n - 2)....3 \times 2 \times 1\]
\[ \Rightarrow \]Number of ways to arrange 6 different flowers \[ = 5 \times 4 \times 3 \times 2 \times 1\]
\[ \Rightarrow \]Number of ways to arrange 6 different flowers \[ = 120\].................… (1)
\[\therefore \]Number of ways to arrange 6 different flowers are 120.
Now we know a garland can be made in two ways, when we move in clockwise direction and when we move in anti-clockwise direction.
Since garland is also circular, we use the formula for circular arrangement here.
So, the number of ways to form a garland will be half the number of ways to arrange flowers in a circle as the number of ways to arrange flowers in a circle contains both arrangements: clockwise and anti-clockwise.
\[ \Rightarrow \]Number of ways to form a garland \[ = \dfrac{1}{2}\] number of ways to arrange flowers in circle
Substitute the value of number of ways to arrange flowers in a circle from equation (1)
\[ \Rightarrow \]Number of ways to form a garland \[ = \dfrac{1}{2} \times 120\]
Cancel same factors from numerator and denominator
 \[ \Rightarrow \]Number of ways to form a garland \[ = 60\]

\[\therefore \]Number of ways to form a garland is 60.

Note: Students many times make the mistake of applying the general formula of permutation here i.e. \[n!\] which is wrong. Keep in mind for arranging any number of items in a circular manner we have the formula \[(n - 1)!\]. Also, garland has no condition here so the sequence of flowers does not matter here.