Ways to form a group from three groups with given constraints

Given three numbers(x, y and z) which denote the number of people in the first group, the second group, and third group. We can form groups by selecting people from the first group, the second group and third group such that the following conditions are not void.

  • A minimum of one people has to be selected from every group.
  • The number of people selected from the first group has to be at least one more than the number of people selected from the third group.

The task is to find the number of ways of forming distinct groups.

Examples:

Input: x = 3, y = 2, z = 1
Output: 9
Lets say x has people (a, b, c)
Y has people (d, e)
Z has people (f)
Then the 9 ways are {a, b, d, f}, {a, b, e, f}, {a, c, d, f}, {a, c, e, f},
{b, c, d, f}, {b, c, e, f}, {a, b, c, d, f}, {a, b, c, e, f} and
{a, b, c, d, e, f}

Input: x = 4, y = 2, z = 1
Output: 27

The problem can be solved using combinatorics. There are three positions(in terms of people from different groups) which needs to be filled. The first has to be filled with a number with one or more greater than the second position. The third can be filled with any number. We know if we need to fill k positions with N people, then the number of ways of doing it is {N \choose K}. Hence the following steps can be followed to solve the above problem.

  • The second position can be filled with i = 1 to i = y people.
  • The first position can be filled with j = i+1 to j = x people.
  • The third position can be filled with any number of k = 1 to k = z people.
  • Hence the common thing is filling the third position with k people. Hence we can take that portion as common.
  • Run two loops( i and j) for filling up the second position and first position respectively.
  • The number of ways of filling up the positions are {y \choose i} * {x \choose j}.
  • After calculation of all ways to fill up those two positions, we can simply multiply summation of {z \choose 1} + {z \choose 2} + … {z \choose z} since that was the common part in both.

{N \choose K} can be pre-computed using Dynamic Programming to reduce time complexity. The method is discussed here.

Below is the implementation of the above approach.

C++

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// C++ program to find the number of
// ways to form the group of peopls
#include <bits/stdc++.h>
using namespace std;
  
int C[1000][1000];
  
// Function to pre-compute the
// Combination using DP
void binomialCoeff(int n)
{
    int i, j;
  
    // Caculate value of Binomial Coefficient
    // in bottom up manner
    for (i = 0; i <= n; i++) {
        for (j = 0; j <= i; j++) {
  
            // Base Cases
            if (j == 0 || j == i)
                C[i][j] = 1;
  
            // Calculate value using previously
            // stored values
            else
                C[i][j] = C[i - 1][j - 1] + C[i - 1][j];
        }
    }
  
    // return C[n][k];
}
  
// Function to find the number of ways
int numberOfWays(int x, int y, int z)
{
    // Function to pre-compute
    binomialCoeff(max(x, max(y, z)));
  
    // Sum the Zci
    int sum = 0;
    for (int i = 1; i <= z; i++) {
        sum = (sum + C[z][i]);
    }
  
    // Iterate for second position
    int sum1 = 0;
    for (int i = 1; i <= y; i++) {
  
        // Iterate for first position
        for (int j = i + 1; j <= x; j++) {
            sum1 = (sum1 + (C[y][i] * C[x][j]));
        }
    }
  
    // Multiply the common Combination value
    sum1 = (sum * sum1);
  
    return sum1;
}
  
// Driver Code
int main()
{
    int x = 3;
    int y = 2;
    int z = 1;
  
    cout << numberOfWays(x, y, z);
  
    return 0;
}

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Java

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// Java program to find the number of 
// ways to form the group of peopls 
class GFG
{
      
static int C[][] = new int [1000][1000]; 
  
// Function to pre-compute the 
// Combination using DP 
static void binomialCoeff(int n) 
    int i, j; 
  
    // Caculate value of Binomial Coefficient 
    // in bottom up manner 
    for (i = 0; i <= n; i++) 
    
        for (j = 0; j <= i; j++) 
        
  
            // Base Cases 
            if (j == 0 || j == i) 
                C[i][j] = 1
  
            // Calculate value using previously 
            // stored values 
            else
                C[i][j] = C[i - 1][j - 1] + C[i - 1][j]; 
        
    
  
    // return C[n][k]; 
  
// Function to find the number of ways 
static int numberOfWays(int x, int y, int z) 
    // Function to pre-compute 
    binomialCoeff(Math.max(x, Math.max(y, z))); 
  
    // Sum the Zci 
    int sum = 0
    for (int i = 1; i <= z; i++) 
    
        sum = (sum + C[z][i]); 
    
  
    // Iterate for second position 
    int sum1 = 0
    for (int i = 1; i <= y; i++)
    
  
        // Iterate for first position 
        for (int j = i + 1; j <= x; j++)
        
            sum1 = (sum1 + (C[y][i] * C[x][j])); 
        
    
  
    // Multiply the common Combination value 
    sum1 = (sum * sum1); 
  
    return sum1; 
  
// Driver Code 
public static void main(String args[]) 
    int x = 3
    int y = 2
    int z = 1
  
    System.out.println(numberOfWays(x, y, z)); 
}
  
// This code is contributed by Arnab Kundu

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C#

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// C# program to find the number of 
// ways to form the group of peopls 
using System;
      
class GFG
{
      
static int [,]C = new int [1000,1000]; 
  
// Function to pre-compute the 
// Combination using DP 
static void binomialCoeff(int n) 
    int i, j; 
  
    // Caculate value of Binomial Coefficient 
    // in bottom up manner 
    for (i = 0; i <= n; i++) 
    
        for (j = 0; j <= i; j++) 
        
  
            // Base Cases 
            if (j == 0 || j == i) 
                C[i,j] = 1; 
  
            // Calculate value using previously 
            // stored values 
            else
                C[i,j] = C[i - 1,j - 1] + C[i - 1,j]; 
        
    
  
    // return C[n,k]; 
  
// Function to find the number of ways 
static int numberOfWays(int x, int y, int z) 
    // Function to pre-compute 
    binomialCoeff(Math.Max(x, Math.Max(y, z))); 
  
    // Sum the Zci 
    int sum = 0; 
    for (int i = 1; i <= z; i++) 
    
        sum = (sum + C[z,i]); 
    
  
    // Iterate for second position 
    int sum1 = 0; 
    for (int i = 1; i <= y; i++)
    
  
        // Iterate for first position 
        for (int j = i + 1; j <= x; j++)
        
            sum1 = (sum1 + (C[y,i] * C[x,j])); 
        
    
  
    // Multiply the common Combination value 
    sum1 = (sum * sum1); 
  
    return sum1; 
  
// Driver Code 
public static void Main(String []args) 
    int x = 3; 
    int y = 2; 
    int z = 1; 
  
    Console.WriteLine(numberOfWays(x, y, z)); 
}
  
// This code is contributed by Rajput-Ji

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Output:

9

Time Complexity: O(K * K ), where K is the max of (x, y and z).



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