In how many ways we can form a garland using 3 different red flowers, 5 different yellow flowers and 4 different blue flowers, if flowers of same colour must be together?

Answer

To solve the problem of forming a garland using 3 different red flowers, 5 different yellow flowers, and 4 different blue flowers, with the condition that flowers of the same color must be together, we can follow these steps:

### Step 1: Treat each color group as a single unit
Since flowers of the same color must be together, we can treat each color group as a single unit. Therefore, we have:
- 1 unit for red flowers (R)
- 1 unit for yellow flowers (Y)
- 1 unit for blue flowers (B)

This gives us a total of 3 units: R, Y, and B.

### Step 2: Arrange the color units
The number of ways to arrange these 3 units in a circular manner (since it’s a garland) is given by the formula for circular permutations, which is \((n - 1)!\) where \(n\) is the number of units. Here, \(n = 3\):
\[ \text{Arrangements of units} = (3 - 1)! = 2! = 2 \]

### Step 3: Arrange the flowers within each color unit
Next, we need to arrange the flowers within each color unit:
- The 3 different red flowers can be arranged in \(3!\) ways.
- The 5 different yellow flowers can be arranged in \(5!\) ways.
- The 4 different blue flowers can be arranged in \(4!\) ways.

Calculating these:
\[ 3! = 6 \]
\[ 5! = 120 \]
\[ 4! = 24 \]

### Step 4: Combine the arrangements
Now, we multiply the number of arrangements of the units by the arrangements of the flowers within each unit:
\[ \text{Total arrangements} = \text{Arrangements of units} \times (3! \times 5! \times 4!) \]
\[ \text{Total arrangements} = 2 \times (6 \times 120 \times 24) \]

### Step 5: Calculate the final result
First, calculate \(6 \times 120 \times 24\):
\[ 6 \times 120 = 720 \]
\[ 720 \times 24 = 17280 \]
Now, multiply by 2:
\[ \text{Total arrangements} = 2 \times 17280 = 34560 \]

Thus, the total number of ways to form the garland is **34,560**.

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