The number of all possible symmetric matrices of order `3xx3` with each entry 1 or 2 and whose sum of diagonal elements is equal to 5, is

Answer

To solve the problem of finding the number of all possible symmetric matrices of order \(3 \times 3\) with each entry being either 1 or 2, and whose sum of diagonal elements equals 5, we can follow these steps:

### Step 1: Understand the structure of a symmetric matrix
A symmetric matrix has the property that \(a_{ij} = a_{ji}\). For a \(3 \times 3\) symmetric matrix, the structure looks like this:

\[ \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{12} & a_{22} & a_{23} \\ a_{13} & a_{23} & a_{33} \end{bmatrix} \]

### Step 2: Set the condition for the diagonal elements
The diagonal elements are \(a_{11}\), \(a_{22}\), and \(a_{33}\). According to the problem, we need:

\[ a_{11} + a_{22} + a_{33} = 5 \]

### Step 3: Determine possible values for diagonal elements
Since each diagonal element can only be 1 or 2, we can find combinations of these values that sum to 5. The possible combinations are:

1. \( (2, 2, 1) \)
2. \( (2, 1, 2) \)
3. \( (1, 2, 2) \)

### Step 4: Count the arrangements of diagonal elements
For each combination, we need to consider the arrangements:

- For \( (2, 2, 1) \), the number of distinct arrangements is given by:

\[ \frac{3!}{2!} = 3 \]

This accounts for the two indistinguishable 2's.

### Step 5: Fill the off-diagonal elements
The off-diagonal elements \(a_{12}\), \(a_{13}\), and \(a_{23}\) can each independently be either 1 or 2. Since there are 3 off-diagonal positions and each can take 2 values (1 or 2), the total number of combinations for these positions is:

\[ 2^3 = 8 \]

### Step 6: Calculate total combinations
Now, for each arrangement of the diagonal elements, we have 8 combinations for the off-diagonal elements. Since there are 3 arrangements of the diagonal elements, the total number of symmetric matrices is:

\[ 3 \times 8 = 24 \]

### Final Answer
Thus, the number of all possible symmetric matrices of order \(3 \times 3\) with each entry being 1 or 2 and whose sum of diagonal elements equals 5 is **24**.

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