Sample solution:
class Program
{
static void Main(string[] args)
{
// The input
var n = 5;
var t = new[] { 2, 4, 5 };
// Helping transformations
ComputeDistances(t);
CorrectDistances(t);
// The algorithm
var r = CalculateRank(t, n);
Console.WriteLine("n = 5");
Console.WriteLine("t = {2, 4, 5}");
Console.WriteLine("r = {0}", r);
Console.ReadKey();
}
static void ComputeDistances(int[] t)
{
var k = t.Length;
while (--k >= 0)
t[k] -= (k + 1);
}
static void CorrectDistances(int[] t)
{
var k = t.Length;
while (--k > 0)
t[k] -= t[k - 1];
}
static int CalculateRank(int[] t, int n)
{
int k = t.Length - 1, r = 0;
for (var i = 0; i < t.Length; i++)
{
if (t[i] == 0)
{
n--;
k--;
continue;
}
for (var j = 0; j < t[i]; j++)
{
n--;
r += CalculateBinomialCoefficient(n, k);
}
n--;
k--;
}
return r;
}
static int CalculateBinomialCoefficient(int n, int k)
{
int i, l = 1, m, x, y;
if (n - k < k)
{
x = k;
y = n - k;
}
else
{
x = n - k;
y = k;
}
for (i = x + 1; i <= n; i++)
l *= i;
m = CalculateFactorial(y);
return l/m;
}
static int CalculateFactorial(int n)
{
int i, w = 1;
for (i = 1; i <= n; i++)
w *= i;
return w;
}
}
The idea behind the scenes is to associate a k-subset with an operation of drawing k-elements from the n-size set. It is a combination, so the overall count of possible items will be (n k). It is a clue that we could seek the solution in Pascal Triangle. After a while of comparing manually written examples with the appropriate numbers from the Pascal Triangle, we will find the pattern and hence the algorithm.
