Question

Let $A =\left[ a _{i j}\right], a _{i j} \in Z \cap[0,4], 1 \leq i, j \leq 2$ The number of matrices $A$ such that the sum of all entries is a prime number $p \in(2,13)$ is _____

Answer 1

As given a+b+c+d=3 or 5 or 7 or 11
if sum=3
(1+x+x2+…..+x4)4→x3
(1−x5)4(1−x)−4→x3
∴4+3−1C3​=6C3​=20
If sum=5
(1−4x5)(1−x)−4→x5
⇒4+5−1C5​−4x4.4+0−1C0​=8C5​−4=52
If sum =7
(1−4x5)(1−x)−4→x7
⇒4+5−1C4​−4.4+0−1C0​=8C5​−4=52
If sum =11
(1−4x5+6x10)(1−x)−4→x11
⇒4+11−1C11​−4⋅4+6−4C6​+6⋅4+1−1C1​
=14C11​−4⋅9C6​+6.4=364−336+24=52
∴ Total matrices =20+52+80+52=204
So, the correct answer is 204.