I was reading a paper called Exact calculation of the number of degrees of freedom of a rigid body constituted by n particles where a method is suggested on how to calculate the degrees of freedom of a n body system in 3 ,4 and higher dimensions. Here three arbitrary particles are selected to form a plane and a vector is fixed perpendicular to the plane. This makes a 3-D coordinate system. Then the number of possible pairs are calculated where the line joining each pair can be regarded as axis of rotation. These (n-3)C2 pairs add to the previous expression of the number of degrees of freedom(which was diverging for large n). This gives the correct value of 6.
My question is: why do they form pairs with only n-3 particles and not n? https://arxiv.org/pdf/1002.2002.pdf
page number 4. Line 11.


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