How many different necklaces can be formed with 6 White and 5 Red beads?

Options

A. 18
B. 24
C. 21
D. 27

Answer

To find the number of different necklaces that can be formed with 6 white beads and 5 red beads, we can follow these steps:

### Step 1: Calculate the Total Number of Beads
First, we need to determine the total number of beads. We have:
- 6 white beads
- 5 red beads

Total beads = 6 + 5 = 11 beads.

**Hint:** Always start by calculating the total number of items you have.

### Step 2: Use the Formula for Arrangements
The formula for the arrangements of n items where there are groups of identical items is given by:
\[ \frac{n!}{n_1! \cdot n_2!} \]
where \( n \) is the total number of items, \( n_1 \) is the number of identical items of one type, and \( n_2 \) is the number of identical items of another type.

In our case:
- \( n = 11 \) (total beads)
- \( n_1 = 6 \) (white beads)
- \( n_2 = 5 \) (red beads)

So, the number of linear arrangements is:
\[ \frac{11!}{6! \cdot 5!} \]

**Hint:** Remember to use factorial notation when calculating arrangements of items.

### Step 3: Adjust for Circular Arrangements
Since we are forming a necklace, we need to adjust for the circular nature of the arrangement. To do this, we divide the number of linear arrangements by the total number of beads (11):
\[ \text{Circular arrangements} = \frac{11!}{6! \cdot 5! \cdot 11} \]

**Hint:** When dealing with circular arrangements, always divide by the number of items to account for rotations.

### Step 4: Account for Flipping
Necklaces can also be flipped, which means we need to divide the result by 2 to account for this symmetry:
\[ \text{Final arrangements} = \frac{11!}{6! \cdot 5! \cdot 11 \cdot 2} \]

**Hint:** Remember that flipping arrangements reduces the total count, so always divide by 2 for symmetrical cases.

### Step 5: Calculate the Result
Now we can compute the value:
1. Calculate \( 11! = 39916800 \)
2. Calculate \( 6! = 720 \)
3. Calculate \( 5! = 120 \)

Now plug these into the formula:
\[ \text{Final arrangements} = \frac{39916800}{720 \cdot 120 \cdot 11 \cdot 2} \]

Calculating the denominator:
- \( 720 \cdot 120 = 86400 \)
- \( 86400 \cdot 11 = 950400 \)
- \( 950400 \cdot 2 = 1900800 \)

Now, divide:
\[ \text{Final arrangements} = \frac{39916800}{1900800} = 21 \]

Thus, the number of different necklaces that can be formed with 6 white and 5 red beads is **21**.

**Final Answer:** 21 (Option C)