Question

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

Answer 1

Answer: 196

As given $a+b+c+d=3$ or 5 or 7 or 11
if $\mathrm{sum}=3$
${\left(1+x+x^2+\dots ..+x^4\right)}^4\to x^3$
${\left(1-x^5\right)}^4(1-x{)}^{-4}\to x^3$
$\therefore {}^{4+3-1}{C}_3={}^6{C}_3=20$
If $\mathrm{sum}=5$
$\left(1-4x^5\right)(1-x{)}^{-4}\to x^5$
$\Rightarrow {}^{4+5-1}{C}_5-4x^{4.4+0-1}{C}_0={}^8{C}_5-4=52$
If sum $=7$
$\left(1-4x^5\right)(1-x{)}^{-4}\to x^7$
$\Rightarrow {}^{4+5-1}{C}_4-{}^{4.4+0-1}{C}_0={}^8{C}_5-4=52$
$\text{ If sum }=11$
$\left(1-4x^5+6x^{10}\right)(1-x{)}^{-4}\to x^{11}$
$\Rightarrow {}^{4+11-1}{C}_{11}-4\cdot {}^{4+6-4}{C}_6+6\cdot {}^{4+1-1}{C}_1$
$={}^{14}{C}_{11}-4\cdot {}^9{C}_6+6.4=364-336+24=52$
$\therefore \text{ Total matrices }=20+52+80+52=204$