Let n be the number obtained on rolling a fair die. If the probability that the system x- n y+z=6 x+( n -2) y+( n +1) z=8 ( n -1) y+z=1 has a unique solution is k 6 , then the sum of k and all possible values of n is :

Step-by-step solution

The coefficient matrix has determinant (A) = -(n-1)(n-2) . The system has a unique solution when (A) 0 , which requires n 1 and n 2 . For a fair die, n 1,2,3,4,5,6 , so the unique solution occurs for n 3,4,5,6 . The probability is 4 6 = 2 3 , giving k = 4 . The sum of k and all possible values of n where unique solution exists is 4 + (3+4+5+6) = 22 .