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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 $........$.
$203$
$202$
$201$
$204$
Solution
As given $a+b+c+d=3$ or $5$ or $7$ or $11$
$\text { if sum }=3$
$\left(1+x+x^2+\ldots++x^4\right)^4 \rightarrow x^3$
$\left(1-x^5\right)^4(1-x)^{-4} \rightarrow x^3$
$\therefore{ }^{4+3-1} C_3={ }^6 C_3=20$
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 $=7$
$\left(1-4 x ^5\right)(1- x )^{-4} \rightarrow x ^7$
$\Rightarrow{ }^{4+5-1} C _4-{ }^{4 \cdot 4+0-1} C _0={ }^8 C _5-4=52$
$\text { If sum }=11$
$\quad\left(1-4 x ^5+6 x ^{10}\right)(1- x )^{-4} \rightarrow x ^{11}$
$\Rightarrow{ }^{4+11-1} C _{11}-4 \cdot{ }^{4+6-4} C _6+6 \cdot{ }^{4+1-1} C _1$
$={ }^{14} C _{11}-4 \cdot{ }^9 C _6+6.4=364-336+24=52$
$\therefore \text { Total matrices }=20+52+80+52=204$
