Number of ways of distributing N identical objects in R distinct groups with no groups empty
Given two integer N and R, the task is to calculate the number of ways to distribute N identical objects into R distinct groups such that no groups are left empty.
Examples:
Input: N = 4, R = 2
Output: 3
No of objects in 1st group = 1, in second group = 3
No of objects in 1st group = 2, in second group = 2
No of objects in 1st group = 3, in second group = 1
Input: N = 5, R = 3
Output: 6
Approach: Idea is to use Multinomial theorem. Let us suppose that x1 objects are placed in the first group, x2 objects are placed in second group and xR objects are placed in the Rth group. It is given that,
x1 + x2 + x3 +…+ xR = N for all xi ? 1 for 1 ? i ? R
Now replace every xi with yi + 1 for all 1 ? i ? R. Now all the y variables are greater than or equal to zero.
The equation becomes,
y1 + y2 + y3 + … + yR + R = N for all yi ? 0 for 1 ? i ? R
y1 + y2 + y3 + … + yR = N – R
It now reduces to that standard multinomial equation whose solution is (N – R) + R – 1CR – 1.
The solution of this equation is given by N – 1CR – 1.
Below is the implementation of the above approach:
C++
#include <bits/stdc++.h>
using namespace std;
int ncr( int n, int r)
{
int ans = 1;
for ( int i = 1; i <= r; i += 1) {
ans *= (n - r + i);
ans /= i;
}
return ans;
}
int NoOfDistributions( int N, int R)
{
return ncr(N - 1, R - 1);
}
int main()
{
int N = 4;
int R = 3;
cout << NoOfDistributions(N, R);
return 0;
}
|
Java
import java.io.*;
class GFG
{
static int ncr( int n, int r)
{
int ans = 1 ;
for ( int i = 1 ; i <= r; i += 1 )
{
ans *= (n - r + i);
ans /= i;
}
return ans;
}
static int NoOfDistributions( int N, int R)
{
return ncr(N - 1 , R - 1 );
}
public static void main (String[] args)
{
int N = 4 ;
int R = 3 ;
System.out.println(NoOfDistributions(N, R));
}
}
|
Python3
def ncr(n, r):
ans = 1
for i in range ( 1 ,r + 1 ):
ans * = (n - r + i)
ans / / = i
return ans
def NoOfDistributions(N, R):
return ncr(N - 1 , R - 1 )
N = 4
R = 3
print (NoOfDistributions(N, R))
|
C#
using System;
class GFG
{
static int ncr( int n, int r)
{
int ans = 1;
for ( int i = 1; i <= r; i += 1)
{
ans *= (n - r + i);
ans /= i;
}
return ans;
}
static int NoOfDistributions( int N, int R)
{
return ncr(N - 1, R - 1);
}
static public void Main ()
{
int N = 4;
int R = 3;
Console.WriteLine(NoOfDistributions(N, R));
}
}
|
Javascript
<script>
function ncr(n, r)
{
let ans = 1;
for (let i = 1; i <= r; i += 1)
{
ans *= (n - r + i);
ans = parseInt(ans / i);
}
return ans;
}
function NoOfDistributions(N, R)
{
return ncr(N - 1, R - 1);
}
let N = 4;
let R = 3;
document.write(NoOfDistributions(N, R));
</script>
|
Time Complexity: O(R)
Auxiliary Space: O(1)
Last Updated :
31 May, 2022
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