Problem 4

Passwords consist of 6 symbols which can be of three types, i.e. letters, special symbols or digits. Assume that there are 30 letters, 20 special symbols and 10 digits. Assume that all such passwords are valid (e.g. A$Y3B4, 00VV7A og ####%%).")

a) How many passwords contain only one type of symbols (for example only special symbols)? Show a formula but you do not have to compute a final answer.

b) Let us now require that a password must contain all three types of symbols,

that is at least one letter, at least one special symbol and at least one digit. How many

passwords fulfill this condition? Show a formula but you do not have to compute a final

answer. Show arguments!

Problem 5

Use mathematical induction to prove the following formula. Argue carefully

and show your work. State the inductive hypothesis and refer to it when it is used!

Show in detail intermediate matrix calculations.

Problem 6

Use mathematical induction to prove the following formula. Argue carefully

and show your work. State the inductive hypothesis and refer to it when it is used.