Count Number of Balanced Permutations - Solution & Explanation

This approach involves generating all distinct permutations of the string and then checking each permutation to determine if it is balanced. For a string of length n, generate all permutations and check if the sum of digits at even indices equals the sum of digits at odd indices.

The code uses Python's itertools.permutations to generate all unique permutations of the input string. For each permutation, it calculates sums of digits at even and odd indices and checks if they are equal. If they are, it counts the permutation as balanced. The final count is returned modulo 10^9 + 7.

The optimized approach involves using backtracking to selectively generate permutations. The algorithm uses a recursive approach for swapping based on the feasibility of forming a balanced sum. It avoids generating all permutations explicitly, thereby reducing the complexity significantly.

The implementation involves backtracking with position tracking. It recursively estimates whether adding a digit at an even or odd position can lead to the required balance by adjusting sums dynamically. A frequency counter is used to ensure digits aren't reused.