Number of necklaces can be formed with 4 identical beads and two distinct jewels is 'k', then k divides

Options

A. `15`
B. `35`
C. `21`
D. `105`

Explanation

To solve the problem of finding the number of necklaces that can be formed with 4 identical beads and 2 distinct jewels, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have 4 identical beads (let's denote them as 'A') and 2 distinct jewels (let's denote them as 'B' and 'C'). We need to find the number of distinct arrangements (necklaces) that can be formed using these beads and jewels.

2. **Using Burnside's Lemma**:
Since necklaces are circular arrangements, we can use Burnside's lemma to count the distinct arrangements considering rotations.

3. **Counting Fixed Arrangements**:
We need to consider how many arrangements remain unchanged under the rotations of the necklace. For a necklace of 6 positions (4 A's and 2 distinct jewels), we can have the following cases based on the number of jewels used:

- **Case 1**: No jewels (only A's).
- Arrangement: AAAA (1 arrangement).

- **Case 2**: 1 jewel (either B or C).
- Arrangements: AABA, AAAB, AABA, etc. (but all are equivalent under rotation).
- Total: 2 arrangements (B and C can be placed in any of the 4 positions, but they are equivalent under rotation).

- **Case 3**: 2 jewels (both B and C).
- Arrangements: AABBC, ABABC, etc.
- We can have the jewels in positions such that they are not adjacent. The arrangements can be calculated as follows:
- Choose 2 positions for B and C from the 4 available, which can be done in \( \binom{4}{2} = 6 \) ways.
- However, we must divide by the number of rotations that yield the same arrangement.

4. **Calculating Total Distinct Arrangements**:
We can summarize the arrangements:
- 1 arrangement for all A's.
- 2 arrangements for 1 jewel.
- 6 arrangements for 2 jewels.

Total arrangements = 1 + 2 + 6 = 9.

5. **Final Calculation**:
The total number of distinct necklaces (k) is 9.

6. **Finding Divisibility**:
The problem states that k divides something. Since k = 9, we can check the divisors of 9, which are 1, 3, and 9.

### Final Answer:
The number of necklaces that can be formed with 4 identical beads and 2 distinct jewels is \( k = 9 \).