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# Find number of ways to form sets from N distinct things with no set of size A or B

• Last Updated : 24 Apr, 2020

Given three numbers N, A, B. The task is to count the number of ways to select things such that there exists no set of size either A or B. Answer can be very large. So, output answer modulo 109+7.

Note: Empty set is not consider as one of the way.

Examples:

Input: N = 4, A = 1, B = 3
Output: 7
Explanation:
The number of ways to form sets of size 2 are 6 (4C2).
The number of ways to form sets of size 4 are 1 (4C4).

Input: N = 10, A = 4, B = 9
Output: 803

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach: The idea is to first find the number of ways including sets of size including A, B and empty sets. Then the remove the number of the sets of size A, B and empty sets.

Below is the implementation of the above approach:

## CPP

 `// C++ program to find number of sets without size A and B``#include ``using` `namespace` `std;``#define mod (int)(1e9 + 7)`` ` `// Function to find a^m1``int` `power(``int` `a, ``int` `m1)``{``    ``if` `(m1 == 0)``        ``return` `1;``    ``else` `if` `(m1 == 1)``        ``return` `a;``    ``else` `if` `(m1 == 2)``        ``return` `(1LL * a * a) % mod;``    ``// If m1 is odd, then return a * a^m1/2 * a^m1/2``    ``else` `if` `(m1 & 1)``        ``return` `(1LL * a * power(power(a, m1 / 2), 2)) % mod;``    ``else``        ``return` `power(power(a, m1 / 2), 2) % mod;``}`` ` `// Function to find factorial of a number``int` `factorial(``int` `x)``{``    ``int` `ans = 1;``    ``for` `(``int` `i = 1; i <= x; i++)``        ``ans = (1LL * ans * i) % mod;`` ` `    ``return` `ans;``}`` ` `// Function to find inverse of x``int` `inverse(``int` `x)``{``    ``return` `power(x, mod - 2);``}`` ` `// Function to find nCr``int` `binomial(``int` `n, ``int` `r)``{``    ``if` `(r > n)``        ``return` `0;`` ` `    ``int` `ans = factorial(n);`` ` `    ``ans = (1LL * ans * inverse(factorial(r))) % mod;`` ` `    ``ans = (1LL * ans * inverse(factorial(n - r))) % mod;`` ` `    ``return` `ans;``}`` ` `// Function to find number of sets without size a and b``int` `number_of_sets(``int` `n, ``int` `a, ``int` `b)``{``    ``// First calculate all sets``    ``int` `ans = power(2, n);`` ` `    ``// Remove sets of size a``    ``ans = ans - binomial(n, a);`` ` `    ``if` `(ans < 0)``        ``ans += mod;`` ` `    ``// Remove sets of size b``    ``ans = ans - binomial(n, b);`` ` `    ``// Remove empty set``    ``ans--;`` ` `    ``if` `(ans < 0)``        ``ans += mod;`` ` `    ``// Return the required answer``    ``return` `ans;``}`` ` `// Driver code``int` `main()``{``    ``int` `N = 4, A = 1, B = 3;`` ` `    ``// Function call``    ``cout << number_of_sets(N, A, B);`` ` `    ``return` `0;``}`

## Java

 `// Java program to find number of sets without size A and B``import` `java.util.*;`` ` `class` `GFG{``static` `final` `int` `mod =(``int``)(1e9 + ``7``);``  ` `// Function to find a^m1``static` `int` `power(``int` `a, ``int` `m1)``{``    ``if` `(m1 == ``0``)``        ``return` `1``;``    ``else` `if` `(m1 == ``1``)``        ``return` `a;``    ``else` `if` `(m1 == ``2``)``        ``return` `(``int``) ((1L * a * a) % mod);``    ``// If m1 is odd, then return a * a^m1/2 * a^m1/2``    ``else` `if` `(m1 % ``2` `== ``1``)``        ``return` `(``int``) ((1L * a * power(power(a, m1 / ``2``), ``2``)) % mod);``    ``else``        ``return` `power(power(a, m1 / ``2``), ``2``) % mod;``}``  ` `// Function to find factorial of a number``static` `int` `factorial(``int` `x)``{``    ``int` `ans = ``1``;``    ``for` `(``int` `i = ``1``; i <= x; i++)``        ``ans = (``int``) ((1L * ans * i) % mod);``  ` `    ``return` `ans;``}``  ` `// Function to find inverse of x``static` `int` `inverse(``int` `x)``{``    ``return` `power(x, mod - ``2``);``}``  ` `// Function to find nCr``static` `int` `binomial(``int` `n, ``int` `r)``{``    ``if` `(r > n)``        ``return` `0``;``  ` `    ``int` `ans = factorial(n);``  ` `    ``ans = (``int``) ((1L * ans * inverse(factorial(r))) % mod);``  ` `    ``ans = (``int``) ((1L * ans * inverse(factorial(n - r))) % mod);``  ` `    ``return` `ans;``}``  ` `// Function to find number of sets without size a and b``static` `int` `number_of_sets(``int` `n, ``int` `a, ``int` `b)``{``    ``// First calculate all sets``    ``int` `ans = power(``2``, n);``  ` `    ``// Remove sets of size a``    ``ans = ans - binomial(n, a);``  ` `    ``if` `(ans < ``0``)``        ``ans += mod;``  ` `    ``// Remove sets of size b``    ``ans = ans - binomial(n, b);``  ` `    ``// Remove empty set``    ``ans--;``  ` `    ``if` `(ans < ``0``)``        ``ans += mod;``  ` `    ``// Return the required answer``    ``return` `ans;``}``  ` `// Driver code``public` `static` `void` `main(String[] args)``{``    ``int` `N = ``4``, A = ``1``, B = ``3``;``  ` `    ``// Function call``    ``System.out.print(number_of_sets(N, A, B));``  ` `}``}`` ` `// This code contributed by sapnasingh4991`

## Python3

 `# Python3 program to find number of ``# sets without size A and B``mod ``=` `10``*``*``9` `+` `7`` ` `# Function to find a^m1``def` `power(a, m1):``    ``if` `(m1 ``=``=` `0``):``        ``return` `1``    ``elif` `(m1 ``=``=` `1``):``        ``return` `a``    ``elif` `(m1 ``=``=` `2``):``        ``return` `(a ``*` `a) ``%` `mod``          ` `    ``# If m1 is odd, then return a * a^m1/2 * a^m1/2``    ``elif` `(m1 & ``1``):``        ``return` `(a ``*` `power(power(a, m1 ``/``/` `2``), ``2``)) ``%` `mod``    ``else``:``        ``return` `power(power(a, m1 ``/``/` `2``), ``2``) ``%` `mod`` ` `# Function to find factorial of a number``def` `factorial(x):``    ``ans ``=` `1``    ``for` `i ``in` `range``(``1``, x ``+` `1``):``        ``ans ``=` `(ans ``*` `i) ``%` `mod`` ` `    ``return` `ans`` ` `# Function to find inverse of x``def` `inverse(x):``    ``return` `power(x, mod ``-` `2``)`` ` `# Function to find nCr``def` `binomial(n, r):``    ``if` `(r > n):``        ``return` `0`` ` `    ``ans ``=` `factorial(n)`` ` `    ``ans ``=` `(ans ``*` `inverse(factorial(r))) ``%` `mod`` ` `    ``ans ``=` `(ans ``*` `inverse(factorial(n ``-` `r))) ``%` `mod`` ` `    ``return` `ans`` ` `# Function to find number of sets without size a and b``def` `number_of_sets(n, a, b):``     ` `    ``# First calculate all sets``    ``ans ``=` `power(``2``, n)`` ` `    ``# Remove sets of size a``    ``ans ``=` `ans ``-` `binomial(n, a)`` ` `    ``if` `(ans < ``0``):``        ``ans ``+``=` `mod`` ` `    ``# Remove sets of size b``    ``ans ``=` `ans ``-` `binomial(n, b)`` ` `    ``# Remove empty set``    ``ans ``-``=` `1`` ` `    ``if` `(ans < ``0``):``        ``ans ``+``=` `mod`` ` `    ``# Return the required answer``    ``return` `ans`` ` `# Driver code``if` `__name__ ``=``=` `'__main__'``:``    ``N ``=` `4``    ``A ``=` `1``    ``B ``=` `3`` ` `    ``# Function call``    ``print``(number_of_sets(N, A, B))`` ` `# This code is contributed by mohit kumar 29    `

## C#

 `// C# program to find number of sets without size A and B``using` `System;`` ` `class` `GFG{``static` `readonly` `int` `mod =(``int``)(1e9 + 7);``   ` `// Function to find a^m1``static` `int` `power(``int` `a, ``int` `m1)``{``    ``if` `(m1 == 0)``        ``return` `1;``    ``else` `if` `(m1 == 1)``        ``return` `a;``    ``else` `if` `(m1 == 2)``        ``return` `(``int``) ((1L * a * a) % mod);``    ``// If m1 is odd, then return a * a^m1/2 * a^m1/2``    ``else` `if` `(m1 % 2 == 1)``        ``return` `(``int``) ((1L * a * power(power(a, m1 / 2), 2)) % mod);``    ``else``        ``return` `power(power(a, m1 / 2), 2) % mod;``}``   ` `// Function to find factorial of a number``static` `int` `factorial(``int` `x)``{``    ``int` `ans = 1;``    ``for` `(``int` `i = 1; i <= x; i++)``        ``ans = (``int``) ((1L * ans * i) % mod);``   ` `    ``return` `ans;``}``   ` `// Function to find inverse of x``static` `int` `inverse(``int` `x)``{``    ``return` `power(x, mod - 2);``}``   ` `// Function to find nCr``static` `int` `binomial(``int` `n, ``int` `r)``{``    ``if` `(r > n)``        ``return` `0;``   ` `    ``int` `ans = factorial(n);``   ` `    ``ans = (``int``) ((1L * ans * inverse(factorial(r))) % mod);``   ` `    ``ans = (``int``) ((1L * ans * inverse(factorial(n - r))) % mod);``   ` `    ``return` `ans;``}``   ` `// Function to find number of sets without size a and b``static` `int` `number_of_sets(``int` `n, ``int` `a, ``int` `b)``{``    ``// First calculate all sets``    ``int` `ans = power(2, n);``   ` `    ``// Remove sets of size a``    ``ans = ans - binomial(n, a);``   ` `    ``if` `(ans < 0)``        ``ans += mod;``   ` `    ``// Remove sets of size b``    ``ans = ans - binomial(n, b);``   ` `    ``// Remove empty set``    ``ans--;``   ` `    ``if` `(ans < 0)``        ``ans += mod;``   ` `    ``// Return the required answer``    ``return` `ans;``}``   ` `// Driver code``public` `static` `void` `Main(String[] args)``{``    ``int` `N = 4, A = 1, B = 3;``   ` `    ``// Function call``    ``Console.Write(number_of_sets(N, A, B));``}``}``  ` `// This code is contributed by PrinciRaj1992`
Output:
```7
```

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