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

Let $A=\begin{bmatrix} a & b \\ c & d \end{bmatrix} ; a , b , c , d \in\{0,1,2,3,4,5\}$ $a+b+c+d=p, p \in\{3,5,7\}$
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 $=180$