In certain mathematician circles, C! is also a shorthand term for "contradiction." The clearest example of its proper usage can be found in a problem involving solving a system of equations.
Suppose that you are given the following system to solve:
(i) x + 4y - z = 4
(ii) 3x - 2y + 5z = 6
(iii) 5x + 6y + 3z = 13
One solution1 to this system involves eliminating one of the variables and then adding or subtracting the resulting equations. Let us add (iii) with 3*(i).
3*(i) 3x + 12y - 3z = 12
(iii) 5x + 6y + 3z = 13
(iv) 8x + 18y = 25
We would then add 5*(i) to (ii) to garner a second equation without a "z" term.
5*(i) 5x + 20y - 5z = 20
(ii) 3x - 2y + 5z = 6
(v) 8x + 18y = 26
Subtracting (iv) from (v) produces the result 0=1 which immediately implies a contradiction. Namely C!. Therefore, a solution with only contradictory solutions C! has no solutions.
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1Other solution methods include but are not limited to the iterative method, back substitution, matrices, and graphs.