XOR maximum-minimum matrices Solution Codechef 2021

 

XOR maximum-minimum matrices Solution Codechef 2021 

SOLUTION

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Chef has an integer matrix A of N rows and M columns, and an integer X.

A submatrix is defined as a rectangle of values inside A. More formally, given two intervals [l,r] and [L,R], the corresponding submatrix consists of the values aij such that l≤i≤r and L≤j≤R. Two submatrices are considered different if one of the ranges [l,r] or [L,R] differs.

Chef uses f(X) to denote the total number of submatrices B in which

max(B)⊕min(B)=X,

where ⊕ denotes bitwise XOR, max(B) denotes the maximum integer in B, and min(B) denotes the minimum integer in B.

Chef does not know in which order he would like to put the rows. He wishes to know the value of f(X) for every permutation of rows, but since the amount of them can be too large, he wants to know the sum of the answers over all permutations.

Input

  • The first line contains three space-separated integers N, M and X.
  • Next N lines contains M integers each: the j-th integer of the i-th line Aij.

Output

Output the sum of f(X) over all permutations of the rows of A.

Constraints

  • 1≤N≤8
  • 1≤M≤105
  • 0≤X,Aij≤109

Sample Input

3 3 3
1 2 1
0 3 0
2 1 2

Sample Output

120

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