Find the number of permutations of n things, taken r at a time, in which 3 particular things :
(i) always occur (ii) will never occur.

Answer

To solve the problem of finding the number of permutations of \( n \) things taken \( r \) at a time with specific conditions on 3 particular things, we can break it down into two cases:

### Case (i): 3 Particular Things Always Occur

1. **Identify the Total Items**: We have \( n \) total items, and we want to find permutations of these items taken \( r \) at a time.

2. **Group the 3 Particular Things**: Since the 3 particular things must always occur, we can treat them as a single unit or block. This means we now have \( n - 3 + 1 = n - 2 \) units to consider (the block of 3 things plus the remaining \( n - 3 \) items).

3. **Determine the Number of Selections**: Since we are taking \( r \) items at a time, and 3 of those are already included in our block, we need to select \( r - 3 \) more items from the remaining \( n - 3 \) items.

4. **Calculate the Permutations**: The number of ways to arrange these \( r \) items, where the block of 3 is treated as one unit, is given by:
\[ P(n-2, r-3) \times 3! \]
Here, \( P(n-2, r-3) \) is the number of permutations of \( n-2 \) items taken \( r-3 \) at a time, and \( 3! \) accounts for the arrangements of the 3 particular items within their block.

### Case (ii): 3 Particular Things Will Never Occur

1. **Identify Remaining Items**: Since the 3 particular things cannot be included in our selection, we only have \( n - 3 \) items left to choose from.

2. **Determine the Number of Selections**: We need to select \( r \) items from these \( n - 3 \) items.

3. **Calculate the Permutations**: The number of ways to arrange \( r \) items from \( n - 3 \) items is given by:
\[ P(n-3, r) \]

### Final Answers

- For case (i): The number of permutations where the 3 particular things always occur is:
\[ P(n-2, r-3) \times 3! \]

- For case (ii): The number of permutations where the 3 particular things never occur is:
\[ P(n-3, r) \]