Count of binary strings of length N having equal count of 0’s and 1’s
Given an integer N, the task is to find the number of binary strings possible of length N having same frequency of 0s and 1s. If such string is possible of length N, print -1.
Note: Since the count can be very large, return the answer modulo 109+7.
Input: N = 2
All possible binary strings of length 2 are “00”, “01”, “10” and “11”.
Among them, “10” and “01” have the same frequency of 0s and 1s.
Hence, the answer is 2.
Strings “0011”, “0101”, “0110”, “1100”, “1010” and “1001” have same frequency of 0s and 1s.
Hence, the answer is 6.
The simplest approach is to generate all possible permutations of a string of length N having equal number of ‘0’ and ‘1’. For every permutation generated, increase the count. Print the total count of permutations generated.
Time Complexity: O(N*N!)
Auxiliary Space: O(N)
The above approach can be optimized by using concepts of Permutation and Combination. Follow the steps below to solve the problem:
- Since N positions need to be filled with equal number of 0‘s and 1‘s, select N/2 positions from the N positions in C(N, N/2) % mod( where mod = 109 + 7) ways to fill with only 1’s.
- Fill the remaining positions in C(N/2, N/2) % mod (i.e 1) way with only 0’s.
Below is the implementation of the above approach:
Time Complexity: O(N2)
Space Complexity: O(N)
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