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"The human hand has 27 degrees of freedom: 4 in each finger,

3 for extension and flexion and one for abduction and

adduction; the thumb is more complicated and has 5 DOF,

leaving 6 DOF for the rotation and translation of the wrist."

(from ElKoura and Singh 2003 'Handrix: Animating the Human Hand' Eurographics/SIGGRAPH Symposium on Computer Animation).

They cite Grant's Atlas of Anatomy.

However, robotics engineers tend to work with reduced DoF models, in particular cutting down on the allowable movements of the fingers and thumb.

Edit:

In fact the human hand has 22 degrees of freedom. I don't see how the wrist has 3 translational degrees as suggested in the previous answer. We would have translational degrees if our limbs could extend/contract in a telescopic manner.

All fingers except the thumb has 3 flexion/extension and 1 abduction/adduction. The thumb is missing one joint therefore that makes a total of 4*4 + 3 = 19 excluding the wrist. The wrist has 3 rotational degrees of freedom, hence 22 DoF in total.

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