If `a_i , i= 1,2,3,4` be four real members of same sign, then the minimum value of ` sum (a_i/a_j) , i , j in {1,2,3,4} , i != j ` is : (a) 6 (b) 8 (c) 12 (d) 24

Options

A. 6
B. 8
C. 12
D. 24

Answer

To solve the problem, we need to find the minimum value of the expression:

\[ S = \sum_{i=1}^{4} \sum_{j=1, j \neq i}^{4} \frac{a_i}{a_j} \]

This expression can be expanded as follows:

1. **Expand the expression**:
\[ S = \frac{a_1}{a_2} + \frac{a_1}{a_3} + \frac{a_1}{a_4} + \frac{a_2}{a_1} + \frac{a_2}{a_3} + \frac{a_2}{a_4} + \frac{a_3}{a_1} + \frac{a_3}{a_2} + \frac{a_3}{a_4} + \frac{a_4}{a_1} + \frac{a_4}{a_2} + \frac{a_4}{a_3} \]

2. **Group the terms**:
Notice that for each pair \( (i, j) \), we have both \( \frac{a_i}{a_j} \) and \( \frac{a_j}{a_i} \). For example, \( \frac{a_1}{a_2} + \frac{a_2}{a_1} \).

3. **Apply the AM-GM inequality**:
For each pair \( \frac{a_i}{a_j} + \frac{a_j}{a_i} \), by the AM-GM inequality:
\[ \frac{a_i}{a_j} + \frac{a_j}{a_i} \geq 2 \]

4. **Count the pairs**:
There are a total of 12 pairs (since there are 4 values of \( a_i \) and each can pair with 3 others). Each pair contributes at least 2 to the sum.

5. **Calculate the minimum value**:
Since there are 6 pairs of the form \( \frac{a_i}{a_j} + \frac{a_j}{a_i} \), the minimum value of \( S \) is:
\[ S \geq 6 \times 2 = 12 \]

Thus, the minimum value of \( S \) is **12**.

### Final Answer:
The minimum value of \( \sum \frac{a_i}{a_j} \) for \( i \neq j \) is **12** (Option C).