Ways to select one or more pairs from two different sets

Given two positive numbers ‘n’ and ‘m’ (n <= m) which represent total number of items of first and second type of sets respectively. Find total number of ways to select at-least one pair by picking one item from first type(I) and another item from second type(II). In any arrangement, an item should not be common between any two pairs.
Note: Since answer can be large, output it in modulo 1000000007.

Input: 2 2
Output: 6
Explanation
Let's denote the items of I type
as a, b and II type as c, d i.e,
Type I -  a, b
Type II - c, d
Ways to arrange one pair at a time
1. a --- c
2. a --- d
3. b --- c
4. b --- d
Ways to arrange two pairs at a time
5. a --- c, b --- d
6. a --- d, b --- c

Input: 2 3
Output: 12

Input: 1 2
Output: 2

The approach is simple, we only need the combination of choosing ‘i‘ items from ‘n‘ type and ‘i‘ items from ‘m‘ type and multiply them(Rule of product) where ‘i‘ varies from 1 to ‘n‘. But we can also permute the resultant product in ‘i’ ways therefore we need to multiply with i!. After that take the sum(Rule of sum) of all resultant product to get the final answer.

\implies\displaystyle \sum_{i=1}^{\text{n}} {}^n\text C_i\cdot{}^m\text C_i \cdot i!
\implies\displaystyle \sum_{i=1}^{\text{n}} \frac{n!}{i!(n-i)!}\cdot \frac{m!}{i!(m-i)!}\cdot i!
\implies\displaystyle \sum_{i=1}^{\text{n}} \frac{n!}{(n-i)!}\cdot \frac{m!}{(m-i)!}\cdot \frac{1}{i!}
\implies\displaystyle\sum_{i=1}^{\text{n}} \frac{{}^n\text P_i\cdot{}^m\text P_i}{i} 

C++

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// C++ program to find total no. of ways
// to form a pair in two different set
#include <bits/stdc++.h>
using namespace std;
  
// initialize global variable so that
// it can access by preCalculate() and
// nCr() function
int* fact, *inverseMod;
const int mod = 1e9 + 7;
  
/* Iterative Function to calculate (x^y)%p in O(log y) */
int power(int x, int y, int p)
{
    int res = 1; // Initialize result
  
    x = x % p; // Update x if it is more than or
               // equal to p
  
    while (y) {
  
        // If y is odd, multiply x with result
        if (y & 1)
            res = (1LL * res * x) % p;
  
        // y must be even now
        y = y >> 1; // y = y/2
        x = (1LL * x * x) % p;
    } // trace(res);
  
    return res;
}
  
// Pre-calculate factorial and
// Inverse of number
void preCalculate(int n)
{
    fact[0] = inverseMod[0] = 1;
    for (int i = 1; i <= n; ++i) {
  
        fact[i] = (1LL * fact[i - 1] * i) % mod;
        inverseMod[i] = power(fact[i], mod - 2, mod);
    }
}
  
// utility function to calculate nCr
int nPr(int a, int b)
{
    return (1LL * fact[a] * inverseMod[a - b]) % mod;
}
  
int countWays(int n, int m)
{
    fact = new int[m + 1];
    inverseMod = new int[m + 1];
  
    // Pre-calculate factorial and
    // inverse of number
    preCalculate(m);
  
    // Initialize answer
    int ans = 0;
    for (int i = 1; i <= n; ++i) {
  
        ans += (1LL * ((1LL * nPr(n, i) 
                * nPr(m, i)) % mod)
                * inverseMod[i]) % mod;
        if (ans >= mod)
            ans %= mod;
    }
  
    return ans;
}
  
// Driver program
int main()
{
    int n = 2, m = 2;
  
    cout << countWays(n, m);
  
    return 0;
}

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Java

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// Java program to find total
// no. of ways to form a pair
// in two different set
  
public class Test
{
    static long[] fact;
    static long[] inverseMod;
    static int mod=1000000007;
      
    /* Iterative Function to calculate 
       (x^y)%p in O(log y) */
      
    static long power(long x, int y, int p)
    {
        // Initialize result
        long res = 1
      
        // Update x if it is more than or
        // equal to p
        x = x % p; 
      
        while (y!=0)
        {
      
            // If y is odd, multiply 
            // x with result
            if ((y & 1)!=0)
                res = (1 * res * x) % p;
            }
            // y must be even now
            y = y >> 1
              
            x = (1 * x * x) % p;
        
      
      
        return res;
    }
      
    // Pre-calculate factorial and
    // Inverse of number
    public static void preCalculate(int n)
    {
        //int fact[]=new long[n];
        // int inverseMod[]=new long[n];
        fact[0] = 1;
        inverseMod[0] = 1;
          
        for (int i = 1; i <= n; i++) 
        {
          
                fact[i] = (1 * fact[i - 1] * i)
                                          % mod;
                                            
                inverseMod[i] = power(fact[i], 
                                  mod - 2, mod);
              
        }
    }
      
    // utility function to calculate nCr
    public static long nPr(int a, int b)
    {
          
        return (1 * fact[a] * inverseMod[a - b])
                                    % (long)mod;
    }
          
    public static int countWays(int n, int m)
    {
          
        fact = new long[m + 1];
        inverseMod = new long[m + 1];
          
        // Pre-calculate factorial and
        // inverse of number
        preCalculate(m);
      
        // Initialize answer
        long ans = 0;
          
        for (int i = 1; i <= n; i++) {
      
            ans = ans+(1 * ((1 * nPr(n, i)* 
                                nPr(m, i)) % mod)*
                                (inverseMod[i])) % mod;
                                            
            if (ans >= mod)
                ans %= mod;
        }
      
        return (int)ans;
     
      
    /* Driver program */
    public static void main(String[] args) 
    {
        int n = 2, m = 2;
          
        System.out.println(countWays(n, m));
    }
}
  
// This code is contributed by Gitanjali

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Output:



6

Time complexity: O(m*log(mod))
Auxiliary space: O(m)

This article is contributed by Shubham Bansal. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.



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