Find the number of ways in which 10 different flowers can be strung to form a garland so that 3 particular flowers are always together .

Options

A. 30240
B. 30420
C. 23400
D. none of (a),(b),(C )

Answer

To solve the problem of finding the number of ways to string 10 different flowers into a garland such that 3 particular flowers are always together, we can follow these steps:

### Step 1: Treat the 3 particular flowers as a single unit
Since the 3 particular flowers must always be together, we can group them into one unit. Let's denote this group as \( S_1 \).

### Step 2: Count the total units
After grouping the 3 flowers into one unit, we now have:
- 1 unit for the group of 3 flowers (\( S_1 \))
- 7 other individual flowers

This gives us a total of \( 1 + 7 = 8 \) units to arrange.

### Step 3: Arrange the units
The number of ways to arrange these 8 units in a linear fashion is given by \( 8! \) (8 factorial).

### Step 4: Arrange the flowers within the group
Within the group \( S_1 \), the 3 flowers can be arranged among themselves. The number of ways to arrange these 3 flowers is \( 3! \) (3 factorial).

### Step 5: Calculate the total arrangements
The total number of arrangements can be calculated by multiplying the arrangements of the units by the arrangements within the group:
\[ \text{Total arrangements} = 8! \times 3! \]

### Step 6: Compute the factorials
Now we compute the values:
- \( 8! = 40320 \)
- \( 3! = 6 \)

### Step 7: Final calculation
Now, multiply these two results:
\[ \text{Total arrangements} = 40320 \times 6 = 241920 \]

Thus, the total number of ways to string the 10 different flowers such that the 3 particular flowers are always together is **241920**.

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