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 = 2Output:4Explanation:

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 = 8Output:5

**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) = r
_{C2}/ (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 = 3andb = 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 <bits/stdc++.h> ` `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; ` `} ` |

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## 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 ` |

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**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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