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 _____ .

Step-by-step solution

Given, A = a 11 a 12 a 21 a 22   &   a 11 , a 12 , a 21 , a 22 ∈ 0 , 1 , 2 , 3 , 4 Now given a 11 + a 12 + a 23 + a 22 = prime number, So sum can be, p = 3 , 5 , 7 , 11 Now using multinomial theorem we get, Sum will be sum of coefficient of x 3 ,   x 5 ,   x 7   &   x 11 in expansion of x 0 + x 1 + x 2 + x 3 + x 4 4 ⇒ 1 - x 5 1 - x 4 = 1 - x 5 4 1 - x - 4 = C r 1 4 - x 5 r 1 C r 2 4 + r 2 - 1 x r 2 = C r 1 3 + r 2 4 C r 2 - 1 r 1 x 5 r 1 + r 2 Now taking, 5 r 1 + r