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# Queries of nCr%p in O(1) time complexity

Given Q queries and P where P is a prime number, each query has two numbers N and R and the task is to calculate nCr mod p.
Constraints:

```N <= 106
R <= 106
p is a prime number```

Examples:

Input:
Q = 2 p = 1000000007
1st query: N = 15, R = 4
2nd query: N = 20, R = 3
Output:
1st query: 1365
2nd query: 1140
15!/(4!*(15-4)!)%1000000007 = 1365
20!/(20!*(20-3)!)%1000000007 = 1140

A naive approach is to calculate nCr using formulae by applying modular operations at any time. Hence time complexity will be O(q*n).

A better approach is to use fermat little theorem. According to it nCr can also be written as (n!/(r!*(n-r)!) ) mod which is equivalent to (n!*inverse(r!)*inverse((n-r)!) ) mod p. So, precomputing factorial of numbers from 1 to n will allow queries to be answered in O(log n). The only calculation that needs to be done is calculating inverse of r! and (n-r)!. Hence overall complexity will be q*( log(n)) .

A efficient approach will be to reduce the better approach to an efficient one by precomputing the inverse of factorials. Precompute inverse of factorial in O(n) time and then queries can be answered in O(1) time. Inverse of 1 to N natural number can be computed in O(n) time using Modular multiplicative inverse. Using recursive definition of factorial, the following can be written:

```n! = n * (n-1) !
taking inverse on both side
inverse( n! ) = inverse( n ) * inverse( (n-1)! )```

Since N’s maximum value is 106, precomputing values till 106 will do.

Below is the implementation of the above approach:

## C++

 `// C++ program to answer queries` `// of nCr in O(1) time.` `#include ` `#define ll long long` `const` `int` `N = 1000001;` `using` `namespace` `std;`   `// array to store inverse of 1 to N` `ll factorialNumInverse[N + 1];`   `// array to precompute inverse of 1! to N!` `ll naturalNumInverse[N + 1];`   `// array to store factorial of first N numbers` `ll fact[N + 1];`   `// Function to precompute inverse of numbers` `void` `InverseofNumber(ll p)` `{` `    ``naturalNumInverse = naturalNumInverse = 1;` `    ``for` `(``int` `i = 2; i <= N; i++)` `        ``naturalNumInverse[i] = naturalNumInverse[p % i] * (p - p / i) % p;` `}` `// Function to precompute inverse of factorials` `void` `InverseofFactorial(ll p)` `{` `    ``factorialNumInverse = factorialNumInverse = 1;`   `    ``// precompute inverse of natural numbers` `    ``for` `(``int` `i = 2; i <= N; i++)` `        ``factorialNumInverse[i] = (naturalNumInverse[i] * factorialNumInverse[i - 1]) % p;` `}`   `// Function to calculate factorial of 1 to N` `void` `factorial(ll p)` `{` `    ``fact = 1;`   `    ``// precompute factorials` `    ``for` `(``int` `i = 1; i <= N; i++) {` `        ``fact[i] = (fact[i - 1] * i) % p;` `    ``}` `}`   `// Function to return nCr % p in O(1) time` `ll Binomial(ll N, ll R, ll p)` `{` `    ``// n C r = n!*inverse(r!)*inverse((n-r)!)` `    ``ll ans = ((fact[N] * factorialNumInverse[R])` `              ``% p * factorialNumInverse[N - R])` `             ``% p;` `    ``return` `ans;` `}`   `// Driver Code` `int` `main()` `{` `    ``// Calling functions to precompute the` `    ``// required arrays which will be required` `    ``// to answer every query in O(1)` `    ``ll p = 1000000007;` `    ``InverseofNumber(p);` `    ``InverseofFactorial(p);` `    ``factorial(p);`   `    ``// 1st query` `    ``ll N = 15;` `    ``ll R = 4;` `    ``cout << Binomial(N, R, p) << endl;`   `    ``// 2nd query` `    ``N = 20;` `    ``R = 3;` `    ``cout << Binomial(N, R, p) << endl;`   `    ``return` `0;` `}`

## Java

 `// Java program to answer queries ` `// of nCr in O(1) time` `import` `java.io.*;`   `class` `GFG{` `    `  `static` `int` `N = ``1000001``;  `   `// Array to store inverse of 1 to N ` `static` `long``[] factorialNumInverse = ``new` `long``[N + ``1``]; `   `// Array to precompute inverse of 1! to N! ` `static` `long``[] naturalNumInverse = ``new` `long``[N + ``1``];`   `// Array to store factorial of first N numbers ` `static` `long``[] fact = ``new` `long``[N + ``1``]; `   `// Function to precompute inverse of numbers ` `public` `static` `void` `InverseofNumber(``int` `p) ` `{ ` `    ``naturalNumInverse[``0``] = naturalNumInverse[``1``] = ``1``; ` `    `  `    ``for``(``int` `i = ``2``; i <= N; i++) ` `        ``naturalNumInverse[i] = naturalNumInverse[p % i] *` `                                 ``(``long``)(p - p / i) % p; ` `} `   `// Function to precompute inverse of factorials ` `public` `static` `void` `InverseofFactorial(``int` `p) ` `{ ` `    ``factorialNumInverse[``0``] = factorialNumInverse[``1``] = ``1``; `   `    ``// Precompute inverse of natural numbers ` `    ``for``(``int` `i = ``2``; i <= N; i++) ` `        ``factorialNumInverse[i] = (naturalNumInverse[i] * ` `                           ``factorialNumInverse[i - ``1``]) % p; ` `} `   `// Function to calculate factorial of 1 to N ` `public` `static` `void` `factorial(``int` `p) ` `{ ` `    ``fact[``0``] = ``1``; `   `    ``// Precompute factorials ` `    ``for``(``int` `i = ``1``; i <= N; i++)` `    ``{ ` `        ``fact[i] = (fact[i - ``1``] * (``long``)i) % p; ` `    ``} ` `} `   `// Function to return nCr % p in O(1) time ` `public` `static` `long` `Binomial(``int` `N, ``int` `R, ``int` `p) ` `{ ` `    `  `    ``// n C r = n!*inverse(r!)*inverse((n-r)!) ` `    ``long` `ans = ((fact[N] * factorialNumInverse[R]) % ` `                       ``p * factorialNumInverse[N - R]) % p; ` `    `  `    ``return` `ans; ` `}`   `// Driver code` `public` `static` `void` `main (String[] args)` `{` `    `  `    ``// Calling functions to precompute the ` `    ``// required arrays which will be required ` `    ``// to answer every query in O(1) ` `    ``int` `p = ``1000000007``; ` `    ``InverseofNumber(p); ` `    ``InverseofFactorial(p); ` `    ``factorial(p); ` `    `  `    ``// 1st query ` `    ``int` `n = ``15``; ` `    ``int` `R = ``4``; ` `    ``System.out.println(Binomial(n, R, p));` `    `  `    ``// 2nd query ` `    ``n = ``20``; ` `    ``R = ``3``; ` `    ``System.out.println(Binomial(n, R, p));` `}` `}`   `// This code is contributed by RohitOberoi`

## Python3

 `# Python3 program to answer queries` `# of nCr in O(1) time.` `N ``=` `1000001`   `# array to store inverse of 1 to N` `factorialNumInverse ``=` `[``None``] ``*` `(N ``+` `1``)`   `# array to precompute inverse of 1! to N!` `naturalNumInverse ``=` `[``None``] ``*` `(N ``+` `1``)`   `# array to store factorial of ` `# first N numbers` `fact ``=` `[``None``] ``*` `(N ``+` `1``)`   `# Function to precompute inverse of numbers` `def` `InverseofNumber(p):` `    ``naturalNumInverse[``0``] ``=` `naturalNumInverse[``1``] ``=` `1` `    ``for` `i ``in` `range``(``2``, N ``+` `1``, ``1``):` `        ``naturalNumInverse[i] ``=` `(naturalNumInverse[p ``%` `i] ``*` `                                   ``(p ``-` `int``(p ``/` `i)) ``%` `p)`   `# Function to precompute inverse ` `# of factorials` `def` `InverseofFactorial(p):` `    ``factorialNumInverse[``0``] ``=` `factorialNumInverse[``1``] ``=` `1`   `    ``# precompute inverse of natural numbers` `    ``for` `i ``in` `range``(``2``, N ``+` `1``, ``1``):` `        ``factorialNumInverse[i] ``=` `(naturalNumInverse[i] ``*` `                                  ``factorialNumInverse[i ``-` `1``]) ``%` `p`   `# Function to calculate factorial of 1 to N` `def` `factorial(p):` `    ``fact[``0``] ``=` `1`   `    ``# precompute factorials` `    ``for` `i ``in` `range``(``1``, N ``+` `1``):` `        ``fact[i] ``=` `(fact[i ``-` `1``] ``*` `i) ``%` `p`   `# Function to return nCr % p in O(1) time` `def` `Binomial(N, R, p):` `    `  `    ``# n C r = n!*inverse(r!)*inverse((n-r)!)` `    ``ans ``=` `((fact[N] ``*` `factorialNumInverse[R])``%` `p ``*` `                      ``factorialNumInverse[N ``-` `R])``%` `p` `    ``return` `ans`   `# Driver Code` `if` `__name__ ``=``=` `'__main__'``:` `    `  `    ``# Calling functions to precompute the` `    ``# required arrays which will be required` `    ``# to answer every query in O(1)` `    ``p ``=` `1000000007` `    ``InverseofNumber(p)` `    ``InverseofFactorial(p)` `    ``factorial(p)`   `    ``# 1st query` `    ``N ``=` `15` `    ``R ``=` `4` `    ``print``(Binomial(N, R, p))`   `    ``# 2nd query` `    ``N ``=` `20` `    ``R ``=` `3` `    ``print``(Binomial(N, R, p))`   `# This code is contributed by` `# Surendra_Gangwar`

## C#

 `// C# program to answer queries  ` `// of nCr in O(1) time ` `using` `System;`   `class` `GFG{` `    `  `static` `int` `N = 1000001;   `   `// Array to store inverse of 1 to N  ` `static` `long``[] factorialNumInverse = ``new` `long``[N + 1];  ` `  `  `// Array to precompute inverse of 1! to N!  ` `static` `long``[] naturalNumInverse = ``new` `long``[N + 1]; ` `  `  `// Array to store factorial of first N numbers  ` `static` `long``[] fact = ``new` `long``[N + 1];  ` `  `  `// Function to precompute inverse of numbers  ` `static` `void` `InverseofNumber(``int` `p)  ` `{  ` `    ``naturalNumInverse = naturalNumInverse = 1;  ` `      `  `    ``for``(``int` `i = 2; i <= N; i++)  ` `        ``naturalNumInverse[i] = naturalNumInverse[p % i] * ` `                                 ``(``long``)(p - p / i) % p;  ` `}  ` `  `  `// Function to precompute inverse of factorials  ` `static` `void` `InverseofFactorial(``int` `p)  ` `{  ` `    ``factorialNumInverse = factorialNumInverse = 1;  ` `  `  `    ``// Precompute inverse of natural numbers  ` `    ``for``(``int` `i = 2; i <= N; i++)  ` `        ``factorialNumInverse[i] = (naturalNumInverse[i] *  ` `                            ``factorialNumInverse[i - 1]) % p;  ` `}  ` `  `  `// Function to calculate factorial of 1 to N  ` `static` `void` `factorial(``int` `p)  ` `{  ` `    ``fact = 1;  ` `  `  `    ``// Precompute factorials  ` `    ``for``(``int` `i = 1; i <= N; i++) ` `    ``{  ` `        ``fact[i] = (fact[i - 1] * (``long``)i) % p;  ` `    ``}  ` `}  ` `  `  `// Function to return nCr % p in O(1) time  ` `static` `long` `Binomial(``int` `N, ``int` `R, ``int` `p)  ` `{  ` `      `  `    ``// n C r = n!*inverse(r!)*inverse((n-r)!)  ` `    ``long` `ans = ((fact[N] * factorialNumInverse[R]) %  ` `                       ``p * factorialNumInverse[N - R]) % p;  ` `      `  `    ``return` `ans;  ` `} `   `// Driver code` `static` `void` `Main() ` `{` `    `  `    ``// Calling functions to precompute the  ` `    ``// required arrays which will be required  ` `    ``// to answer every query in O(1)  ` `    ``int` `p = 1000000007;  ` `    ``InverseofNumber(p);  ` `    ``InverseofFactorial(p);  ` `    ``factorial(p);  ` `      `  `    ``// 1st query  ` `    ``int` `n = 15;  ` `    ``int` `R = 4;  ` `    ``Console.WriteLine(Binomial(n, R, p));` `      `  `    ``// 2nd query  ` `    ``n = 20;  ` `    ``R = 3;  ` `    ``Console.WriteLine(Binomial(n, R, p));` `}` `}`   `// This code is contributed by divyeshrabadiya07`

## Javascript

 ``

Output:

```1365
1140```

Time Complexity: O(N) for precomputing and O(1) for every query.
Auxiliary Space: O(N)

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