The number of ways that a garland can be made out of 6 red and 5 white roses so that no two white roses come together is

Options

A. 21600
B. 43200
C. 40320
D. 5040

Answer

To solve the problem of how many ways a garland can be made using 6 red and 5 white roses such that no two white roses are adjacent, we can follow these steps:

### Step 1: Arranging the Red Roses
First, we arrange the 6 red roses in a circular manner. When arranging n items in a circle, the formula used is (n - 1)!. Therefore, for 6 red roses, the number of arrangements is:

\[ (6 - 1)! = 5! = 120 \]

### Step 2: Identifying Spaces for White Roses
Once the red roses are arranged, they create spaces where the white roses can be placed. In a circular arrangement of 6 red roses, there will be 6 gaps (spaces) available for placing the white roses (one gap between each pair of red roses).

### Step 3: Placing the White Roses
We need to place 5 white roses in these 6 gaps, ensuring that no two white roses are adjacent. Since we have 6 gaps and we need to choose 5 of them to place the white roses, we can use the combination formula:

\[ \text{Number of ways to choose 5 gaps from 6} = \binom{6}{5} = 6 \]

### Step 4: Total Arrangements
Now, we multiply the number of arrangements of red roses by the number of ways to place the white roses:

\[ \text{Total arrangements} = \text{Arrangements of red roses} \times \text{Ways to place white roses} \]
\[ = 120 \times 6 = 720 \]

### Final Answer
Thus, the total number of ways to make the garland with the given conditions is:

\[ \boxed{720} \]

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