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 $\operatorname{sum}=3$
$ \left(1+x+x^2+\ldots . .+x^4\right)^4 \rightarrow x^3 $
$ \left(1-x^5\right)^4(1-x)^{-4} \rightarrow x^3 $
$ \therefore{ }^{4+3-1} C_3={ }^6 C_3=20$
If $\operatorname{sum}=5$
$ \left(1-4 x ^5\right)(1- x )^{-4} \rightarrow x ^5 $
$\Rightarrow{ }^{4+5-1} C _5-4 x ^{4.4+0-1} C _0={ }^8 C _5-4=52$
If sum $=7$
$ \left(1-4 x ^5\right)(1- x )^{-4} \rightarrow x ^7 $
$ \Rightarrow{ }^{4+5-1} C _4-{ }^{4.4+0-1} C _0={ }^8 C _5-4=52 $
$ \text { If sum }=11 $
$ \qquad \left(1-4 x ^5+6 x ^{10}\right)(1- x )^{-4} \rightarrow x ^{11} $
$ \Rightarrow{ }^{4+11-1} C _{11}-4 \cdot{ }^{4+6-4} C _6+6 \cdot{ }^{4+1-1} C _1 $
$ ={ }^{14} C _{11}-4 \cdot{ }^9 C _6+6.4=364-336+24=52 $
$ \therefore \text { Total matrices }=20+52+80+52=204$