The number of matrices of order $$3 \times 3$$, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is _______

We need to count $$3 \times 3$$ matrices with entries 0 or 1 such that the sum of all 9 entries is a prime number. Each entry is independently 0 or 1, so the sum of entries ranges from 0 to 9.

The prime values in $$\{0, 1, 2, \ldots, 9\}$$ are $$\{2, 3, 5, 7\}$$.

The number of matrices with entry sum equal to $$k$$ is $$\binom{9}{k}$$ (choosing which $$k$$ of the 9 entries are 1). So the total count is:

$$\binom{9}{2} + \binom{9}{3} + \binom{9}{5} + \binom{9}{7}$$

We compute each: $$\binom{9}{2} = 36$$, $$\binom{9}{3} = 84$$, $$\binom{9}{5} = 126$$, $$\binom{9}{7} = 36$$.

The total is $$36 + 84 + 126 + 36 = 282$$.

Hence, the correct answer is 282.