Find the number of permutations of `n` distinct things taken `r` together, in which 3 particular things must occur together.

Explanation

To solve the problem of finding the number of permutations of `n` distinct things taken `r` together, with the condition that 3 particular things must occur together, we can follow these steps:

### Step-by-Step Solution:

1. **Group the 3 Particular Things**:
Since the 3 particular things must occur together, we can treat them as a single unit or block. This reduces the problem to arranging this block along with the remaining `n - 3` distinct things.

2. **Calculate the Effective Count of Things**:
After grouping the 3 particular things, we have:
- 1 block (the group of 3 things)
- `n - 3` other distinct things
Hence, the total number of units to arrange is:
\[ 1 + (n - 3) = n - 2 \]

3. **Determine the Number of Selections**:
We need to select `r` items from these `n - 2` units. However, since we have already grouped the 3 particular things, we need to ensure that we account for the selection of these units correctly. The number of ways to choose `r - 1` units (since one unit is the block of 3) from the remaining `n - 3` units is given by:
\[ \binom{n - 3}{r - 3} \]

4. **Arrange the 3 Particular Things**:
The 3 particular things can be arranged among themselves in:
\[ 3! \text{ (factorial of 3)} \]

5. **Arrange All Units**:
The total number of ways to arrange the `r - 2` units (which includes the block of 3 as one unit) is:
\[ (r - 2)! \]

6. **Combine All Parts**:
Therefore, the total number of permutations of `n` distinct things taken `r` together, where 3 particular things must occur together, is given by:
\[ \text{Total permutations} = \binom{n - 3}{r - 3} \times 3! \times (r - 2)! \]

### Final Answer:
Thus, the number of permutations of `n` distinct things taken `r` together, in which 3 particular things must occur together, is:
\[ \binom{n - 3}{r - 3} \times 3! \times (r - 2)! \]

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