Consider 23 different coloured beads in a neclace. In how many ways can the beads be placed in the necklace so that 3 specific beads always remain together?

Answer

To solve the problem of arranging 23 different colored beads in a necklace such that 3 specific beads always remain together, we can follow these steps:

### Step-by-Step Solution:

1. **Treat the 3 specific beads as a single unit**: Since we want the 3 specific beads to always remain together, we can consider them as one single unit or "block". This means instead of arranging 23 individual beads, we will arrange 21 units (the block of 3 beads + the remaining 20 beads).

**Hint**: Think of the 3 beads as one "super bead" to simplify the arrangement.

2. **Calculate the total number of units**: We have 20 individual beads plus 1 block of 3 beads, giving us a total of 21 units to arrange.

**Hint**: Count the total number of items you are arranging after grouping.

3. **Use the formula for circular permutations**: When arranging items in a circle, the formula for the number of arrangements is (n-1)!, where n is the total number of units. Here, n = 21, so we calculate (21 - 1)! = 20!.

**Hint**: Remember that in circular arrangements, one position is fixed to avoid counting rotations as different arrangements.

4. **Account for the arrangements within the block**: The 3 specific beads can be arranged among themselves in 3! (factorial) ways. Therefore, we need to multiply the circular arrangements by the arrangements of the 3 beads.

**Hint**: Factor in the internal arrangements of the grouped items.

5. **Combine the results**: The total number of arrangements is given by:
\[ \text{Total arrangements} = 20! \times 3! \]

6. **Calculate the final answer**:
\[ 3! = 6 \quad \text{(since } 3! = 3 \times 2 \times 1\text{)} \]
Therefore, the total number of arrangements is:
\[ 20! \times 6 \]

### Final Answer:
The total number of ways to arrange the beads in the necklace such that the 3 specific beads remain together is \( 20! \times 6 \).