# Maximum sum of a Matrix where each value is from a unique row and column

Given a matrix of size N X N, the task is to find maximum sum of this Matrix where each value picked is from a unique column for every row.

Examples:

```Input: matrix = [[3, 4, 4, 4],
[1, 3, 4, 4],
[3, 2, 3, 4],
[4, 4, 4, 4]]
Output: 16
Explanation:
Selecting (0, 1) from row 1 = 4
Selecting (1, 2) from row 2 = 4
Selecting (2, 3) from row 3 = 4
Selecting (3, 0) from row 4 = 4
Therefore, max sum = 4 + 4 + 4 + 4 = 16

Input: matrix = [[0, 1, 0, 1],
[3, 0, 0, 2],
[1, 0, 2, 0],
[0, 2, 0, 0]]
Output: 8
Explanation:
Selecting (0, 3) from row 1 = 1
Selecting (1, 0) from row 2 = 3
Selecting (2, 2) from row 3 = 2
Selecting (3, 1) from row 4 = 2
Therefore, max sum = 1 + 3 + 2 + 2 = 8
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach:

• Genrate a numeric string of size N containing numbers from 1 to N
• Find the permutation of this string (N!).
• Now pairing is done between the permutations, such that each N! pairing has a unique column for every row.
• Then calculate the sum of values for all the pairs.

Below is the implementation of the above approach:

 `# Python code for maximum sum of  ` `# a Matrix where each value is  ` `# from a unique row and column ` ` `  `# Permutations using library function ` `from` `itertools ``import` `permutations ` ` `  `# Function MaxSum to find ` `# maximum sum in matrix ` `def` `MaxSum(side, matrix): ` `         `  `    ``s ``=` `'' ` `    ``# Generating the string  ` `    ``for` `i ``in` `range``(``0``, side): ` `        ``s ``=` `s ``+` `str``(i)  ` ` `  `    ``# Permutations of s string ` `    ``permlist ``=` `permutations(s) ` `     `  `    ``# List all possible case ` `    ``cases ``=` `[] ` `     `  `    ``# Append all possible case in cases list ` `    ``for` `perm ``in` `list``(permlist): ` `        ``cases.append(''.join(perm)) ` `     `  `    ``# List to store all Sums  ` `    ``sum` `=` `[] ` `     `  `    ``# Iterate over all case ` `    ``for` `c ``in` `cases: ` `        ``p ``=` `[] ` `        ``tot ``=` `0` `        ``for` `i ``in` `range``(``0``, side): ` `            ``p.append(matrix[``int``(s[i])][``int``(c[i])]) ` `        ``p.sort() ` `        ``for` `i ``in` `range``(side``-``1``, ``-``1``, ``-``1``): ` `            ``tot ``=` `tot ``+` `p[i] ` `        ``sum``.append(tot) ` `     `  ` `  `    ``# Maximum of sum list is  ` `    ``# the max sum ` `    ``print``(``max``(``sum``))  ` ` `  `         `  `# Driver code  ` `if` `__name__ ``=``=` `'__main__'``:  ` `     `  `    ``side ``=` `4` `    ``matrix ``=` `[[``3``, ``4``, ``4``, ``4``], [``1``, ``3``, ``4``, ``4``], [``3``, ``2``, ``3``, ``4``], [``4``, ``4``, ``4``, ``4``]] ` `    ``MaxSum(side, matrix) ` `    ``side ``=` `3` `    ``matrix ``=` `[[``1``, ``2``, ``3``], [``6``, ``5``, ``4``], [``7``, ``9``, ``8``]] ` `    ``MaxSum(side, matrix) `

Output:

```16
18
```

Time complexity: O(K), where K = N!

Auxiliary Space complexity: O(K), where K = N!

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