Let $A = [a_{ij}]$, $a_{ij} \in \mathbb{Z} \cap [0,4]$, $1 \leq i, j \leq 2$. The number of matrices A such that the sum of all entries is a prime number $p \in (2,13)$ is ___.
Step-by-Step Solution
Key Concept: Primes in $(2,13)$ are 3, 5, 7, 11; count 4-tuples $(a_{11},a_{12},a_{21},a_{22})$ from $\{0,1,2,3,4\}$ with sum equal to each prime and add
Primes in $(2,13)$: 3, 5, 7, 11. Count ways to write each as sum of four values from $\{0,...,4\}$ using generating functions or case analysis. The counts are: sum=3: 20, sum=5: 56, sum=7: 80, sum=11: 40. Total = 196. Answer: 196
Correct Answer: 196