If one quarter of all three element subsete of the set `A={a_(1),a_(2),a_(3),......,a_(n)}` contains the element `a_(3)`, then n=

Options

A. 10
B. 12
C. 14
D. none of these

Explanation

To solve the problem, we need to find the value of \( n \) given that one quarter of all three-element subsets of the set \( A = \{a_1, a_2, a_3, \ldots, a_n\} \) contains the element \( a_3 \).

### Step-by-Step Solution:

1. **Determine the Total Number of Three-Element Subsets:**
The total number of three-element subsets of a set with \( n \) elements is given by the combination formula:
\[
\binom{n}{3} = \frac{n(n-1)(n-2)}{3!} = \frac{n(n-1)(n-2)}{6}
...