float to int conversion going wrong (even though the float is already an int)

I was writing a little function to calculate the binomial coefficiant using the tgamma function provided by c++. tgamma returns float values, but I wanted to return an integer. Please take a look at this example program comparing three ways of converting the float back to an int:

#include <iostream>
#include <cmath>

int BinCoeffnear(int n,int k){
    return std::nearbyint( std::tgamma(n+1) / (std::tgamma(k+1)*std::tgamma(n-k+1)) );
}

int BinCoeffcast(int n,int k){
    return static_cast<int>( std::tgamma(n+1) / (std::tgamma(k+1)*std::tgamma(n-k+1)) );
}

int BinCoeff(int n,int k){
    return (int) std::tgamma(n+1) / (std::tgamma(k+1)*std::tgamma(n-k+1));
}

int main()
{
    int n = 7;
    int k = 2;
    std::cout << "Correct: " << std::tgamma(7+1) / (std::tgamma(2+1)*std::tgamma(7-2+1)); //returns 21
    std::cout << " BinCoeff: " << BinCoeff(n,k); //returns 20 
    std::cout << " StaticCast: " << BinCoeffcast(n,k); //returns 20
    std::cout << " nearby int: " << BinCoeffnear(n,k); //returns 21

    return 0;   
}

why is it, that even though the calculation returns a float equal to 21, 'normal' conversion fails and only nearbyint returns the correct value. What is the nicest way to implement this?

EDIT: according to c++ documentation here tgamma(int) should return a double...