Let A = [ a i j ] , a i j Z [ 0 , 4 ] , 1 i , j 2 . The number of matrices A such that the sum of all entries is a prime number p ( 2 , 13 ) is ________ . [2023]

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Solution

Q.
Let $A = [a_{i j}], a_{i j} \in Z \cap [0, 4], 1 \leq i, j \leq 2$ . The number of matrices A such that the sum of all entries is a prime number $p \in (2, 13)$ is ________ . [2023]

Ans.
(204)

As given $a + b + c + d = 3 or 5 or 7 or 11$

If sum = 3, then

Coefficient of $x^{3}$ in ${(1 + x + x^{2} + \dots + x^{4})}^{4}$ is ${(1 - x^{5})}^{4} {(1 - x)}^{- 4}$

$∴$ ${}^{4 + 3 - 1}{C_{3} = {}^{6}{C_{3}}} = 20$

If sum = 5, coefficient of $x^{5}$ is $(1 - 4 x^{5}) {(1 - x)}^{- 4}$

$∴$ ${}^{4 + 5 - 1}{C_{5}} - 4 = {}^{8}{C_{5}} - 4 = 52$

If sum = 7, then coefficient of $x^{7}$ is

$(1 - 4 x^{5}) {(1 - x)}^{- 4} \Rightarrow {}^{10}C_{7} - 4 \times {}^{5}{C_{2}} = 80$

If sum = 11

$(1 - 4 x^{5} + 6 x^{10}) {(1 - x)}^{- 4}$ $\Rightarrow {}^{4 + 11 - 1}{C_{11}} - 4 {}^{4 + 6 - 1}{C_{6}} + 6 {}^{4 + 1 - 1}{C_{1}}$

$= {}^{14}{C_{11}} - 4$ ${}^{9}{C_{6}} + 24 = 364 - 336 + 24 = 52$

$∴ Total matrices = 20 + 52 + 80 + 52 = 204$

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