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Assume you want to design an algorithm that auto-generates a 5 character password

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Assume you want to design an algorithm that auto-generates a 5 character password. You determine that the algorithm will only choose from the following characters:

• 5 uppercase vowels: A, E, I, O, U
• 5 lowercase vowels: a, e, i, o, u
• 5 special symbols: !, @, #, $, %
• 10 integers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Remember that the algorithm will only choose from the characters listed above.

(a) Assume the algorithm chooses randomly from the entire list of possibilities listed above for each character (with the possibility of repeating a character). How many unique 5 character passwords are possible? 

(b) Find the probability that the algorithm generates a 5 character password with only two integers where the first and the fifth characters are integers and are not the same. (i.e, the 5 character password begins and ends with two different integers and the other characters are non-integers) 

(c) Find the probability that the algorithm generates a 5 character password with exactly two special symbols (with repeat characters allowed). 

(d) Find the probability that at least one of the characters in the 5 characters password is either an upper or lowercase vowel. 

(e) A hacker trying to break one of your 5 character passwords runs an algorithm that accurately identifies the first 2 characters of the password before being stopped. However, the hacker's algorithm has determined that the last 3 characters have exactly one special symbol, one lower case letter, and one integer. What is the probability the hacker successfully finds the correct password?