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

JEE Main · 2023 · 31 Jan 2023 · Shift 2 · Permutations and Combinations

Let

A = [a_{ij}], a_{ij} \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: 204

Each of the four entries lies in

{0, 1, 2, 3, 4}

, and their sum must be a prime in

(2, 13)

, namely

3, 5, 7, 11.

The count of ordered quadruples summing to

k

is the coefficient of

x^{k}

(1 + x + x^{2} + x^{3} + x^{4})^{4}

: for

k = 3

it is

20

, for

k = 5

it is

56 - 4 = 52

, for

k = 7

it is

120 - 40 = 80

, and for

k = 11

(symmetric to

5

) it is

52.

The total is

20 + 52 + 80 + 52 = 204.

JEE Main · 2023 · 31 Jan 2023 · Shift 2 · Integer answer