I am trying to understand the degree of freedom of a rigid body.
See: The body has N particles, totalizing a degree of freedom of 3N. BUT, there are constraints such that $|r_{i}-r_{j}|^2 - C_{ij}=0$.
So i just thought that we could calculate the degree of freedom by $3N - C_{N,2}$. The combination arise because is the way to combinate the r's. But this is wrong, the degree of freedom should be always 6.
I can see why it need to be 6, but why we didn't get 6 in $3N - C_{N,2}$? Is this equation wrong? Why?