# Minimum size binary string required such that probability of deleting two 1’s at random is 1/X

Given a value X, the task is to find a minimum size binary string, such that if any 2 characters are deleted at random, the probability that both the characters will be ‘1’ is 1/X. Print the size of such binary string.
Example:

Input: X = 2
Output:
Explanation:
Let the binary string be “0111”.
Probability of choosing 2 1s from given string is = 3C2 / 4C2 = 3/6 = 1/2 (which is equal to 1/X).
Hence the required size is 4.
(Any 4 size binary string with 3 ‘1’s and 1 ‘0’ can be taken for this example).
Input: X = 8
Output:

Approach: We will try to find a formula to solve this problem.

Let
r = Number of 1’s in the string
and
b = Number of 0’s in the string.

• If two characters are deleted at random, then

Total number of ways = (r + b) C 2.

• If 2 characters are desired to be 1’s, Favourable number of cases = r C 2

• Hence, P(both are 1’s) = rC2 / (r + b)C2.
• A tricky observation to further proceed our calculation is:
• Squaring the inequality and comparing with the equality, we get
• If r > 1, we take square root on all 3 sides.
• Taking the leftmost part of the inequality, we get:

• Similarly, taking the rightmost part of the inequality, we get:
• Combining the derived conclusions, we get the range of r in terms of b.
• For the minimum value of string, we set b = 1
• In order to get a valid minimum r, we take the first integer value of r in this range.

For Example: if X = 2

Hence, r = 3 and b = 1
P(both character are 1’s) = 3C2 / 4C2 = 2/4 = 1/2

Below is the implementation of the above approach.

## C++

 // C++ implementation of the  // above approach     #include  using namespace std;     // Function returns the minimum  // size of the string  int MinimumString(int x)  {      // From formula      int b = 1;         // Left limit of r      double left_lim = sqrt(x) + 1.0;         // Right limit of r      double right_lim = sqrt(x) + 2.0;         int r;      for (int i = left_lim; i <= right_lim; i++) {          if (i > left_lim and i < right_lim) {              // Smallest integer in              // the valid range              r = i;              break;          }      }         return b + r;  }     // Driver Code  int main()  {         int X = 2;      cout << MinimumString(X);      return 0;  }

## Python3

 # Python3 implementation of  # the above approach  from math import sqrt     # Function returns the minimum  # size of the string  def MinimumString(x):         # From formula      b = 1        # Left limit of r      left_lim = sqrt(x) + 1.0        # Right limit of r      right_lim = sqrt(x) + 2.0        for i in range(int(left_lim),                      int(right_lim) + 1):          if(i > left_lim and i < right_lim):                             # Smallest integer in              # the valid range               r = i              break        return b + r     # Driver Code  if __name__ == '__main__':         X = 2        print(MinimumString(X))     # This code is contributed by Shivam Singh

Output:

4


Time Complexity: O(1), as the difference between left_lim and right_lim will be always less than 1.
Auxiliary Space: O(1)

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