# Permutation (nPr) and Combination (nCr) in C

Permutation is the number of ways an ordered subset can be obtained of $r$ elements from a set of $n$ elements.

The below example illustrates the number of ways in which **three** different circles (red, yellow and black) can be permutated, taken **two** at a time. It is 6.

The permutation of $n$ things taken $r$ at a time is denoted by $_{n} \text{P}^{r}$ and the formula for obtaining it is

$_{n} \text{P}^{r} = \frac{n!}{(n-r)!}$

Below we write a C program to compute the permutation of $n$ things taken $r$ at a time, based on the above formula. A little note to be taken here, your input should be $r \leq n$.

` ````
#include <stdio.h>
unsigned long nPr(unsigned short n, unsigned short r);
unsigned long factorial(unsigned short int);
int main() {
unsigned short int n, r;
printf("Enter n: ");
scanf("%hu", &n);
printf("Enter r: ");
scanf("%hu", &r);
printf("%huP%hu: %lu", n, r, nPr(n,r));
printf("\n");
return 0;
}
unsigned long nPr(unsigned short n, unsigned short r) {
return factorial(n)/factorial(n-r);
}
unsigned long factorial(unsigned short n) {
unsigned long f;
if(n == 0 || n == 1)
return 1;
else
f = n * factorial(n-1);
return f;
}
```

We run the program for $n = 6$ and $r = 2$

` ````
$ ./a.out
Enter n: 6
Enter r: 2
```

It outputs:

` ````
6P2: 30
```

When order is not considered, it becomes a **combination**. The below example illustrates the number of ways in which **three** different circles (red, yellow and black) can be combined, taken **two** at a time. It is 3.

The combination of $n$ things taken $r$ at a time is denoted by $_{n} \text{C}^{r}$ and the formula for obtaining it is

$_{n} \text{C}^{r} = \frac{n!}{r!(n-r)!}$

It is also known as the Binomial Coefficient. Below is the C program based on the above formula.

` ````
#include <stdio.h>
unsigned long nCr(unsigned short n, unsigned short r);
unsigned long factorial(unsigned short int);
int main() {
unsigned short int n, r;
printf("Enter n: ");
scanf("%hu", &n);
printf("Enter r: ");
scanf("%hu", &r);
printf("%huC%hu: %lu", n, r, nCr(n,r));
printf("\n");
return 0;
}
unsigned long nCr(unsigned short n, unsigned short r) {
return factorial(n)/(factorial(r)*factorial(n-r));
}
unsigned long factorial(unsigned short n) {
unsigned long f;
if(n == 0 || n == 1)
return 1;
else
f = n * factorial(n-1);
return f;
}
```

We execute the above program for $n = 10$ and $r = 6$

` ````
$ ./a.out
Enter n: 10
Enter r: 6
```

And we get the output:

` ````
10C6: 210
```