C++ Program to Find nPr and nCr

In this article, you will learn and get code to find and print ⁿP_r (permutation) and ⁿC_r (combination) in the C++ language. Here is the list of programs available in this article:

Using a while loop, find nPr
Using a while loop, find nCr
Find nPr and nCr in a single program
Using a for loop, calculate nPr and nCr
Find nPr and nCr using a user-defined function

Before going through these programs, let's understand the formula used to calculate permutation (nPr) and combination (nCr).

nPr Formula

The formula to find the value of nPr is:

nPr = n!/(n-r)!

Note: The symbol (!) indicates factorial.

Note: The permutation (nPr) value shows the number of ways to arrange r things out of n.

nCr Formula

The formula to find the value of nCr is:

nCr = n!/r!(n-r)!

Note: The combination (nCr) value shows the number of ways to select r things out of n.

C++: Find nPr (permutation)

To find nPr in C++ programming, you have to ask the user to enter the values of n and r. In both the program for calculating nPr and nCr, the only required code is to find the factorial of a number. The rest of the things are just normal things like initializing, dividing, printing, etc.

The question is, "Write a program in C++ to find the value of nPr." Here is its answer:

#include<iostream>
using namespace std;
int main()
{
    long long fact=1, numerator, denominator;
    int perm, n, r, i=1, sub;
    cout<<"Enter the Value of n: ";
    cin>>n;
    cout<<"Enter the Value of r: ";
    cin>>r;
    while(i<=n)
    {
        fact = i*fact;
        i++;
    }
    numerator = fact;                    // n!
    sub = n-r;
    fact = 1;
    i = 1;
    while(i<=sub)
    {
        fact = i*fact;
        i++;
    }
    denominator = fact;                // (n-r)!
    perm = numerator/denominator;
    cout<<"\nPermutation (nPr) = "<<perm;
    cout<<endl;
    return 0;
}

This program was built and runs under the Code::Blocks IDE. Here is the sample run:

Now type a number, say 5, as the value of n, and then type another number, say 3, as the value of r. Now press the ENTER key to find and print the value of nPr as shown in the output given below:

There are 60 ways to arrange three things out of five.

The following block of code:

while(i<=n)
{
    fact = i*fact;
    i++;
}

is used to find the factorial of n. So the statement:

numerator=fact;

is used to initialized the factorial of n to the numerator. In other words, numerator = n!

In the above program, the factorial of n and n-r gets calculated, and then using the formula of nPr, its value gets calculated and printed on the output. For more information, consult the factorial program.

C++: Find nCr (combination)

This program asks the user to enter the value of n and r. Based on the values, the program finds and prints the value of nCr, which tells the number of ways to select r things out of n:

#include<iostream>
using namespace std;
int main()
{
    long long fact=1, numerator, denominator;
    int comb, n, r, i=1, sub;
    cout<<"Enter the Value of n: ";
    cin>>n;
    cout<<"Enter the Value of r: ";
    cin>>r;
    while(i<=n)
    {
        fact = i*fact;
        i++;
    }
    numerator = fact;                    // n!
    sub = n-r;
    fact = 1;
    i = 1;
    while(i<=sub)
    {
        fact = i*fact;
        i++;
    }
    denominator = fact;               // (n-r)!
    fact = 1;
    i = 1;
    while(i<=r)
    {
        fact = i*fact;
        i++;
    }
    denominator = (fact*denominator);
    comb = numerator/denominator;
    cout<<"\nCombination (nCr) = "<<comb;
    cout<<endl;
    return 0;
}

Here is its sample run with user input, with 5 as the value of n and 3 as the value of r:

There are 10 ways to select 3 things out of 5.

Find nPr and nCr in a Single C++ Program

This program receives the values of n and r. Then it calculates and prints the permutation and combination values for both.

#include<iostream>
using namespace std;
int main()
{
    long long fact=1, numerator, denominator;
    int npr, ncr, n, r, i=1, sub;
    cout<<"Enter the Value of n: ";
    cin>>n;
    cout<<"Enter the Value of r: ";
    cin>>r;
    while(i<=n)
    {
        fact = i*fact;
        i++;
    }
    numerator=fact;                   // fact = n!
    sub = n-r;
    fact=1, i=1;
    while(i<=sub)
    {
        fact = i*fact;
        i++;
    }
    denominator = fact;               // fact = (n-r)!
    npr = numerator/denominator;
    i=1, fact=1;
    while(i<=r)
    {
        fact = i*fact;
        i++;
    }
    ncr = npr/fact;                   // fact = r!
    cout<<"\nPermutation (nPr) = "<<npr;
    cout<<"\nCombination (nCr) = "<<ncr;
    cout<<endl;
    return 0;
}

Here is its sample run with the same user input as provided in the above program's sample run:

Using a for loop, calculate nPr and nCr

This program uses a for loop (instead of while) to do the same job as the previous program.

#include<iostream>
using namespace std;
int main()
{
    long long fact, numerator, denominator;
    int npr, ncr, n, r, i, sub;
    cout<<"Enter the Value of n: ";
    cin>>n;
    cout<<"Enter the Value of r: ";
    cin>>r;
    for(i=1, fact=1; i<=n; i++)
        fact = i*fact;
    numerator=fact;                   // fact = n!
    sub = n-r;
    for(i=1, fact=1; i<=sub; i++)
        fact = i*fact;
    denominator = fact;               // fact = (n-r)!
    npr = numerator/denominator;
    for(i=1, fact=1; i<=r; i++)
        fact = i*fact;
    ncr = npr/fact;                   // fact = r!
    cout<<"\nPermutation (nPr) = "<<npr;
    cout<<"\nCombination (nCr) = "<<ncr;
    cout<<endl;
    return 0;
}

This program produces the same output as the previous program.

Using Function, calculate nPr and nCr.

This is the last program of this article and uses some user-defined functions to do the same task as the previous program.

#include<iostream>
using namespace std;
long long findFact(int);
int findNPR(int, int);
int findNCR(int, int);
int main()
{
    int npr, ncr, n, r;
    cout<<"Enter the Value of n: ";
    cin>>n;
    cout<<"Enter the Value of r: ";
    cin>>r;
    npr = findNPR(n, r);
    ncr = findNCR(n, r);
    cout<<"\nPermutation (nPr) = "<<npr;
    cout<<"\nCombination (nCr) = "<<ncr;
    cout<<endl;
    return 0;
}
long long findFact(int num)
{
    int i=1, fact=1;
    while(i<=num)
    {
        fact = i*fact;
        i++;
    }
    return fact;
}
int findNPR(int n, int r)
{
    long long numerator, denominator;
    numerator = findFact(n);
    denominator = findFact(n-r);
    return (numerator/denominator);
}
int findNCR(int n, int r)
{
    int npr, ncr;
    npr = findNPR(n, r);
    ncr = npr/findFact(r);
    return ncr;
}