Count the corners, edges and faces of a cut cube
Perl 6, 59 58 bytes



{6+@_,|((2,3 X*4+@_)X-(@_ X~@_)∩~&lt;&lt;ords "&#31;% 286
&#14;&#20;&gt;C/")}


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Uses the numbers 0 to 7 to represent the corners. I probably should have matched them up to the same order as in the question... oops? Outputs a list in the order faces, corners, edges.