Question

Let $A$ be a matrix of order $2\times 2$, whose entries are from the set $\{0,1,2,3,4,5\}$. If the sum of all the entries of $A$ is a prime number $p,2<p<8$, then the number of such matrices $A$ is :

Answer 1

Answer: 180

Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right];a,b,c,d\in \{0,1,2,3,4,5\}$ $a+b+c+d=p,p\in \{3,5,7\}$
Case-(i)
$a+b+c+d=3;a,b,c,d\in \{0,1,2,3\}$
No. of ways $={}^{3+4-1}{C}_{4-1}={}^6{C}_3=56$ ...(1)
Case-(ii)
$a+b+c+d=5;a,b,c,d\in \{0,1,2,3,4,5\}$
No. of ways $={}^{3+4-1}{C}_{4-1}={}^8{C}_3=56$ ...(2)
Case-(iii)
$a+b+c+d=7$
No. of ways $=$ total ways when $a,b,c,d\in \{0,1,2$, $3,4,5,6,7\}$ - total ways when a, b, c, d $\notin \{6,7\}$
No of ways $={}^{7+4-1}{C}_{4-1}=\left(\frac{\lfloor 4}{\lfloor 3}+\frac{4}{\underline{2}}\right)$
$={}^{10}{C}_3-16=104$
Hence total no. of ways $=180$