 A set $S$ is called [i]neighbouring [/i] if it has the following two properties:
a)	$S$ has exactly four elements
b)	for every element $x$ of $S$, at least one of the numbers $x - 1$ or $x+1$ belongs to $S$.
Find the number of all [i]neighbouring [/i] subsets of the set $\{1,2,... ,n\}$.
1. **Define the set \( S \) and its properties:**
   - \( S \) is a set with exactly four elements.
   - For every element \( x \) in \( S \), at least one of the numbers \( x-1 \) or \( x+1 \) must also be in \( S \).

2. **Rearrange the elements of \( S \) in ascending order:**
   - Let \( S = \{a_1, a_2, a_3, a_4\} \) where \( a_1 < a_2 < a_3 < a_4 \).

3. **Analyze the neighboring property:**
   - For the property to hold, \( \{a_1, a_2\} \) and \( \{a_3, a_4\} \) must be consecutive pairs.
   - This implies \( a_2 = a_1 + 1 \) and \( a_4 = a_3 + 1 \).

4. **Determine the possible values for \( a_2 \) and \( a_3 \):**
   - Since \( a_1 = a_2 - 1 \) and \( a_4 = a_3 + 1 \), \( a_2 \) and \( a_3 \) must be chosen from the set \( \{2, 3, \ldots, n-1\} \).
   - \( a_2 \) and \( a_3 \) must satisfy \( a_2 < a_3 \).

5. **Count the number of ways to choose \( a_2 \) and \( a_3 \):**
   - The number of ways to choose 2 elements from \( \{2, 3, \ldots, n-1\} \) is given by the binomial coefficient \( \binom{n-2}{2} \).

6. **Conclusion:**
   - The number of neighboring subsets of the set \( \{1, 2, \ldots, n\} \) is \( \boxed{\binom{n-2}{2}} \).