Find the number of all three elements subsets of the set`{a_1, a_2, a_3, a_n}` which contain `a_3dot`

Explanation

To solve the problem of finding the number of all three-element subsets of the set {a_1, a_2, a_3, ..., a_n} that contain the element a_3, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Fixed Element**: Since the subset must contain the element a_3, we can consider a_3 as a fixed element in our subset.

2. **Count Remaining Elements**: After fixing a_3, we are left with the remaining elements of the set, which are {a_1, a_2, a_4, a_5, ..., a_n}. This means we have (n - 1) elements left to choose from.

3. **Determine the Number of Elements to Choose**: Since we need a total of three elements in our subset and one of them is already chosen (a_3), we need to select 2 more elements from the remaining (n - 1) elements.

4. **Use the Combination Formula**: The number of ways to choose 2 elements from (n - 1) elements can be calculated using the combination formula:
\[ \binom{n-1}{2} \]
where \(\binom{n-1}{2}\) is the number of combinations of (n - 1) elements taken 2 at a time.

5. **Final Answer**: Therefore, the number of all three-element subsets of the set {a_1, a_2, a_3, ..., a_n} that include the element a_3 is given by:
\[ \binom{n-1}{2} \]