# 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