How many necklace of 11 beads each can be made from 23 beads of various colours ?

Options

A. `1/(22)((23!)/(12!))`
B. `1/(22)((23!)/(12!))`
C. `((23!)/(2.12!))`
D. `23((21!)/(12!))`

Answer

To solve the problem of how many necklaces of 11 beads can be made from 23 beads of various colors, we will follow these steps:

### Step 1: Choose 11 beads from 23
We first need to determine how many ways we can choose 11 beads from a total of 23 beads. This can be calculated using the combination formula:

\[ \text{Number of ways to choose 11 beads} = \binom{23}{11} = \frac{23!}{11! \cdot (23 - 11)!} = \frac{23!}{11! \cdot 12!} \]

### Step 2: Arrange the 11 beads in a necklace
Next, we need to arrange these 11 beads in a circular manner. The number of arrangements of n objects in a circle is given by \((n - 1)!\). Therefore, for 11 beads, the number of arrangements is:

\[ \text{Number of arrangements in a circle} = (11 - 1)! = 10! \]

### Step 3: Account for symmetrical arrangements
Since a necklace can be flipped (i.e., it can be viewed from two orientations: clockwise and counterclockwise), we need to divide the total arrangements by 2 to account for this symmetry:

\[ \text{Adjusted arrangements} = \frac{10!}{2} \]

### Step 4: Combine the results
Now, we can find the total number of unique necklaces by multiplying the number of ways to choose the beads by the adjusted arrangements:

\[ \text{Total necklaces} = \binom{23}{11} \times \frac{10!}{2} \]

### Step 5: Substitute the values
Substituting the values we calculated:

\[ \text{Total necklaces} = \frac{23!}{11! \cdot 12!} \times \frac{10!}{2} \]

### Final Result
This gives us the total number of unique necklaces that can be made from 23 beads of various colors.