Final answer:
The number of symmetric matrices of order 3 with entries from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} is 360,000.
Step-by-step explanation:
In order to find the number of symmetric matrices of order 3 with entries from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, we need to consider each element of the matrix.
For the main diagonal elements of the matrix, there are 10 choices for each element, so there are a total of 10 * 10 * 10 = 1000 combinations.
For the off-diagonal elements, since the matrix is symmetric, each element below the main diagonal is determined by the corresponding element above the main diagonal. That means, we only have to consider half of the elements. For each off-diagonal element, there are 10 choices, so there are a total of 10 * 9 * 8 / 2 = 360 combinations.
Therefore, the total number of symmetric matrices of order 3 with entries from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} is 1000 * 360 = 360,000.