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

To solve the problem of arranging 6 different flowers in a circle and forming a garland using these flowers, we can follow these steps:

### Step 1: Understand the Circular Arrangement
When arranging items in a circle, we need to account for the fact that rotations of the same arrangement are considered identical. Therefore, the formula to find the number of arrangements of n distinct objects in a circle is given by:

\[ (n-1)! \]

### Step 2: Apply the Formula for Circular Arrangement
For our case, we have 6 different flowers. Using the formula:

\[ \text{Number of arrangements in a circle} = (6-1)! = 5! \]

### Step 3: Calculate 5!
Now, we need to calculate \(5!\):

\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

### Step 4: Consider the Garland Arrangement
When forming a garland, we also need to consider that arrangements can be flipped (i.e., clockwise and anti-clockwise arrangements are the same). Therefore, we divide the total arrangements by 2:

\[ \text{Number of ways to form a garland} = \frac{(n-1)!}{2} = \frac{5!}{2} \]

### Step 5: Calculate the Number of Ways to Form a Garland
Substituting \(5!\) into the equation:

\[ \text{Number of ways to form a garland} = \frac{120}{2} = 60 \]

### Final Answer
Thus, the number of ways to arrange 6 different flowers in a circle is 120, and the number of ways to form a garland using these flowers is 60.

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