You can use this code to get the answer , I will upload an explanation how to solve this using combinatorics soon ,

include <stdio.h>

int main ()

{

int k =1,a,b,c,d,counter=0;

for (k=1;k<=27272;k++)

{ 

    int i=k;

    a=(i % 10);

    i-=a;

    i=i/10;

    b=(i % 10);

    i-=b;

    i=i/10;

    c=(i % 10);

    i-=c;

    i=i/10;

    d=(i % 10);

    i-=d;

    i=i/10;



    if((a+b+c+d+i)==27)

    {

        counter++;

    }

}

printf("counter=%d",counter);

return 0;

}

The output is $666$