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

Answer

To find the number of permutations of \( n \) distinct things taken \( r \) together, where 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. Let's denote this block as \( X \).

2. **Determine the Effective Count of Items**:
After grouping the 3 particular things into one block, we now have:
- 1 block (the group of 3 things)
- \( n - 3 \) other distinct items
Thus, the total number of items we are now considering is \( (n - 3) + 1 = n - 2 \).

3. **Adjust the Selection Count**:
Since we are taking \( r \) items together and we have already counted the block of 3 as one item, we need to adjust our selection count. The number of items we can select from is now \( n - 2 \), and we need to select \( r - 3 \) additional items (since the block of 3 counts as one).

4. **Calculate the Number of Ways to Choose**:
The number of ways to choose \( r - 3 \) items from \( n - 2 \) items is given by the combination formula:
\[ \binom{n - 2}{r - 3} \]

5. **Permutations of the Selected Items**:
After choosing \( r - 3 \) items, we have \( r - 2 \) items in total (the block of 3 plus the \( r - 3 \) selected items). The number of ways to arrange these \( r - 2 \) items is given by:
\[ (r - 2)! \]

6. **Permutations of the Block**:
The 3 particular things within the block can be arranged among themselves in \( 3! \) ways.

7. **Total Permutations**:
Therefore, the total number of permutations where the 3 particular things occur together is:
\[ \text{Total permutations} = \binom{n - 2}{r - 3} \times (r - 2)! \times 3! \]

### Final Formula
So, the final formula for the number of permutations of \( n \) distinct things taken \( r \) together, where 3 particular things must occur together, is:
\[ \text{Total permutations} = \binom{n - 2}{r - 3} \times (r - 2)! \times 3! \]