Count Number of Happy Strings A happy string is:

*

*6 characters long

*has exactly one digit 0-9

*has exactly one special character from:! @ # $ % ^

*the rest are lowercase letters a-z

*it does not contain two of the same letters

Count the number of happy strings that can be made.
My initial thought is $10\cdot6\cdot26\cdot25\cdot24\cdot23$ but I'm afraid I don't take into account the order these characters appear (especially where the digit and special character go).
 A: Your analysis is good, but lacks a factor of $(6 \times 5)$ which represents that there are $6$ positions where the number can go, and then $5$ remaining positions where the character can go.
So, the final computation is in fact
$$(6 \times 5) \times 10 \times 6 \times \frac{(26)!}{(22)!}.$$
The rightmost fraction above, represents :

*

*the number of ways of selecting the $4$ alphabetic characters


*sampling without replacement


*where order of selection is deemed relevant


*and where the positions that the $4$ alphabetic characters will be placed is fixed.   
The positions of the $4$ alphabetic characters are fixed, once the position of the numeric character and the special character are determined.
My answer agrees with your analysis exactly, except for the $(6 \times 5)$ factor that you omitted.
A: First choose your characters, then order them:

*

*${{10}\choose 1}$ for selecting one digit


*${6\choose 1}$ for selecting one special digit


*${{26}\choose 4}$ for selecting four different lowercase letters
Therefore, you have ${{10}\choose 1}{6\choose 1}{{26}\choose 4}$ for choosing the characters as a set.
Since you want to also order them you multiply this answer by $6!$.
