Crypto Algorithm Question - Shift Cipher- No. of non-Fixed Mapping substitution tables possible.

A substitution table is said to be a fixed mapping if at least one of the characters maps to itself, i.e. A-->A.
How many substitution tables will be there where not even a single alphabet maps to itself.

My solution:-

I first tried to find out number of substitution tables with fixed matching.
If only 26 alphabet maps to itself- No. of substitution tables with fixed mapping will be 1
If only 25 alphabets maps to itself- No. of substitution tables with fixed mapping will be 0
If only 24 alphabets maps to itself- No. of substitution tables with fixed mapping will be (26C2)
If only 23 alphabets maps to itself- No. of substitution tables with fixed mapping will be (26C3)*2
If only 22 alphabets maps to itself- No. of substitution tables with fixed mapping will be (26C4)*9
If only 22 alphabets maps to itself- No. of substitution tables with fixed mapping will be (26C5)*44

The main problem is that there is not a general formula for calculating no of fixed mapping if we know that a particular no. of alphabets map to themselves.

If you have any other solution/approach to this problem, please share it.

Rewrite your post using more precise definitions.