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Count of all possible pairs of disjoint subsets of integers from 1 to N
• Last Updated : 30 Aug, 2019

Given an integer N. Consider the set of first N natural numbers A = {1, 2, 3, …, N}. Let M and P be two non-empty subsets of A. The task is to count the number of unordered pairs of (M, P) such that M and P are disjoint sets. Note that the order of M and P doesn’t matter.

Examples:

Input: N = 3
Output: 6
The unordered pairs are ({1}, {2}), ({1}, {3}),
({2}, {3}), ({1}, {2, 3}), ({2}, {1, 3}), ({3}, {1, 2}).

Input: N = 2
Output: 1

Input: N = 10
Output: 28501

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

Approach:

1. Lets assume there are only 6 elements in the set {1, 2, 3, 4, 5, 6}.
2. When you count the number of subsets with 1 as one of the element of first subset, it comes out to be 211.
3. Counting number of subsets with 2 being one of the element of first subset, it comes out to be 65, because 1’s not included as order of sets doesn’t matter.
4. Counting number of subset with 3 being one of the element of first set it comes out to be 65, here a pattern can be observed.
5. Pattern:

5 = 3 * 1 + 2
19 = 3 * 5 + 4
65 = 3 * 19 + 8
211 = 3 * 65 + 16
S(n) = 3 * S(n-1) + 2(n – 2)

6. Expanding it until n->2 (means numbers of elements n-2+1=n-1)
2(n-2) * 3(0) + 2(n – 3) * 31 + 2(n – 4) * 32 + 2(n – 5) * 33 + … + 2(0) * 3(n – 2)
From Geometric progression, a + a * r0 + a * r1 + … + a * r(n – 1) = a * (rn – 1) / (r – 1)
7. S(n) = 3(n – 1) – 2(n – 1). Remember S(n) is number of subsets with 1 as a one of the elements of first subset but to get the required result, Denoted by T(n) = S(1) + S(2) + S(3) + … +S(n)
8. It also forms a Geometric progression, so we calculate it by formula of sum of GP
T(n) = (3n – 2n + 1 + 1)/2
9. As we require T(n) % p where p = 109 + 7
We have to use Fermats’s little theorem
a-1 = a(m – 2) (mod m) for modular division

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach`` ` `#include ``using` `namespace` `std;``#define p 1000000007`` ` `// Modulo exponentiation function``long` `long` `power(``long` `long` `x, ``long` `long` `y)``{``    ``// Function to calculate (x^y)%p in O(log(y))``    ``long` `long` `res = 1;``    ``x = x % p;`` ` `    ``while` `(y > 0) {``        ``if` `(y & 1)``            ``res = (res * x) % p;``        ``y = y >> 1;``        ``x = (x * x) % p;``    ``}`` ` `    ``return` `res % p;``}`` ` `// Driver function``int` `main()``{``    ``long` `long` `n = 3;`` ` `    ``// Evaluating ((3^n-2^(n+1)+1)/2)%p``    ``long` `long` `x = (power(3, n) % p + 1) % p;`` ` `    ``x = (x - power(2, n + 1) + p) % p;`` ` `    ``// From  Fermats’s little theorem``    ``// a^-1 ? a^(m-2) (mod m)`` ` `    ``x = (x * power(2, p - 2)) % p;``    ``cout << x << ``"\n"``;``}`

## Java

 `// Java implementation of the approach``class` `GFG ``{ ``static` `int` `p = ``1000000007``;`` ` `// Modulo exponentiation function``static` `long` `power(``long` `x, ``long` `y)``{``    ``// Function to calculate (x^y)%p in O(log(y))``    ``long` `res = ``1``;``    ``x = x % p;`` ` `    ``while` `(y > ``0``)``    ``{``        ``if` `(y % ``2` `== ``1``)``            ``res = (res * x) % p;``        ``y = y >> ``1``;``        ``x = (x * x) % p;``    ``}``    ``return` `res % p;``}`` ` `// Driver Code``public` `static` `void` `main(String[] args) ``{``    ``long` `n = ``3``;`` ` `    ``// Evaluating ((3^n-2^(n+1)+1)/2)%p``    ``long` `x = (power(``3``, n) % p + ``1``) % p;`` ` `    ``x = (x - power(``2``, n + ``1``) + p) % p;`` ` `    ``// From Fermats's little theorem``    ``// a^-1 ? a^(m-2) (mod m)`` ` `    ``x = (x * power(``2``, p - ``2``)) % p;``    ``System.out.println(x);``}``}`` ` `// This code is contributed by Rajput-Ji`

## Python3

 `# Python3 implementation of the approach``p ``=` `1000000007`` ` `# Modulo exponentiation function``def` `power(x, y):``     ` `    ``# Function to calculate (x^y)%p in O(log(y))``    ``res ``=` `1``    ``x ``=` `x ``%` `p`` ` `    ``while` `(y > ``0``):``        ``if` `(y & ``1``):``            ``res ``=` `(res ``*` `x) ``%` `p;``        ``y ``=` `y >> ``1``        ``x ``=` `(x ``*` `x) ``%` `p`` ` `    ``return` `res ``%` `p`` ` `# Driver Code``n ``=` `3`` ` `# Evaluating ((3^n-2^(n+1)+1)/2)%p``x ``=` `(power(``3``, n) ``%` `p ``+` `1``) ``%` `p`` ` `x ``=` `(x ``-` `power(``2``, n ``+` `1``) ``+` `p) ``%` `p`` ` `# From Fermats’s little theorem``# a^-1 ? a^(m-2) (mod m)``x ``=` `(x ``*` `power(``2``, p ``-` `2``)) ``%` `p`` ` `print``(x)`` ` `# This code is contributed by Mohit Kumar`

## C#

 `// C# implementation of the approach``using` `System;`` ` `class` `GFG``{``static` `int` `p = 1000000007;`` ` `// Modulo exponentiation function``static` `long` `power(``long` `x, ``long` `y)``{``    ``// Function to calculate (x^y)%p in O(log(y))``    ``long` `res = 1;``    ``x = x % p;`` ` `    ``while` `(y > 0)``    ``{``        ``if` `(y % 2 == 1)``            ``res = (res * x) % p;``        ``y = y >> 1;``        ``x = (x * x) % p;``    ``}``    ``return` `res % p;``}`` ` `// Driver Code``static` `public` `void` `Main ()``{``    ``long` `n = 3;``     ` `    ``// Evaluating ((3^n-2^(n+1)+1)/2)%p``    ``long` `x = (power(3, n) % p + 1) % p;``     ` `    ``x = (x - power(2, n + 1) + p) % p;``     ` `    ``// From Fermats's little theorem``    ``// a^-1 ? a^(m-2) (mod m)``     ` `    ``x = (x * power(2, p - 2)) % p;``    ``Console.Write(x);``}``}`` ` `// This code is contributed by ajit.`
Output:
```6
```

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