Assume that a legal password is $10$ characters long, using the following $3$ types of symbols: $26$ alphabets, $10$ digits and $6$ special characters (such as ! and *). In addition, it must use at least $2$ of these types of symbols. How many legal passwords are there? Repeat the previous question with the additional condition that using only digits and special characters is not allowed.
I believe it's a permutation with repetition. Because of the restriction on the first part of the equation, I want to take out all cases where only one type of character is used. So, my answer would be $42^{10} - 26^{10} - 10^{10} - 6^{10}$. For the second part of the question, I think the number of cases where only digits and special characters are used is $16^{10}$ so I would just subtract that from my original answer.
Your answer to the first question is correct.
In the second question, you subtracted the passwords consisting only of digits and those consisting only of special characters twice. So the correct answer is $42^{10}-26^{10}-16^{10}$.