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 the sum of the diagonal elements equal to 5, we can follow these steps:

### Step 1: Identify the diagonal elements
A symmetric \(3 \times 3\) matrix has the following structure:

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

Here, \(a_{11}\), \(a_{22}\), and \(a_{33}\) are the diagonal elements. We need to find combinations of these diagonal elements such that their sum is equal to 5.

### Step 2: Determine possible values for diagonal elements
Since each diagonal element can only be 1 or 2, we can denote the number of 1's as \(x\) and the number of 2's as \(y\). The equation governing the diagonal elements is:

\[ x + 2y = 5 \]

Given that \(x + y = 3\) (since there are three diagonal elements), we can solve these equations.

### Step 3: Solve for combinations of \(x\) and \(y\)
From \(x + y = 3\), we can express \(y\) in terms of \(x\):

\[ y = 3 - x \]

Substituting into the first equation:

\[ x + 2(3 - x) = 5 \\ x + 6 - 2x = 5 \\ -x + 6 = 5 \\ -x = -1 \\ x = 1 \]

Now substituting \(x = 1\) back to find \(y\):

\[ y = 3 - 1 = 2 \]

Thus, we have one combination: \(x = 1\) (one 1) and \(y = 2\) (two 2's).

### Step 4: Count the arrangements of diagonal elements
The diagonal elements can be arranged as follows: we have one 1 and two 2's. The number of distinct arrangements of these elements is given by the multinomial coefficient:

\[ \frac{3!}{1! \cdot 2!} = \frac{6}{1 \cdot 2} = 3 \]

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

\[ 2^3 = 8 \]

### Step 6: Combine the results
Now, to find the total number of symmetric matrices, we multiply the number of arrangements of the diagonal elements by the number of combinations of the off-diagonal elements:

\[ \text{Total Matrices} = \text{Arrangements of Diagonal} \times \text{Combinations of Off-diagonal} = 3 \times 8 = 24 \]

Thus, the total number of all possible symmetric matrices of order \(3 \times 3\) with each entry being 1 or 2 and the sum of diagonal elements equal to 5 is:

\[ \boxed{24} \]