Count the corners, edges and faces of a cut cube Coming from this sandbox post
This is inspired from an 8th graders math test
Intro
We have a cube with following corners
A(0, 0, 0)
B(1, 0, 0)
C(1, 1, 0)
D(0, 1, 0)
E(0, 0, 1)
F(1, 0, 1)
G(1, 1, 1)
H(0, 1, 1)

This cube clearly has 8 corners, 12 edges and 6 faces. If we now cut off corner G, such that our cut plane goes exactly through the middle of each adjacent original edge, we add 2 new corners, 3 new edges and one new face.  Please enjoy this hand drawn piece of art, for better clarification

Input
Given a list of corners (in this example identified by A-H), that will  be cut off, compute the new number of corners, edges and faces.
You make take the input in any form you like, as long as it responds to the same corners (e.g. instead of A-H you can use 1-8 or 0-7, you can assume it to be a list, csv, whatever)
You can assume the list to be distinct (every corner will appear once at most), but it may be empty. The list will never contain non existing corners.
Output
Output three numbers corresponding to the number of corners, edges and faces. Output as a list is explicitly allowed.
Trailing whitespaces are allowed
Examples
{}        -&gt;  8, 12,  6 (empty list)  
{A}       -&gt; 10, 15,  7  
{A,C}     -&gt; 12, 18,  8   
{A,C,F}   -&gt; 14, 21,  9  
{A,B,C}   -&gt; 12, 19,  9  
{A,B,C,D} -&gt; 12, 20, 10  

Finally, this is codegolf, thus the shortest answer in bytes wins. Please refrain from standard loopholes.
Perl 6, 59 58 bytes



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


Try it online!

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.