A secret code for a bank vault consists of 2 letters, then 4 digits and then 3 more letters. How many different codes are possible? How many codes are possible if repeating letters and digits is not allowed? How many codes are possible if repeats are not allowed and the first letter must be 'X' and the first digit must be '4'? If the wrong code is entered the vault automatically locks and the alarm sounds. Suppose repeating letters and digits are allowed in the code. What is ther probability of a thief breaking into the vault if the thief has no prior knowledge of the secret code?

Answer and Step-by-step explanation:

Code: 2 letters | 4 digits | 3 letters

Options for letters: 26 (a b c d e f g h i j k l m n o p q r s t u v w x y z)

Options for letters: 10 (0 1 2 3 4 5 6 7 8 9)

How many different codes are possible?

26.26.10.10.10.10.26.26.26 = 26⁵.10⁴

How many codes are possible if repeating letters and digits is not allowed?

26.25.10.9.8.7.24.23.22 = 3.97837.10¹⁰

How many codes are possible if repeats are not allowed and the first letter must be 'X' and the first digit must be '4'?

1.25.23.1.9.8.7.24.23.22 = 1.53014.10⁸

If the wrong code is entered the vault automatically locks and the alarm sounds. Suppose repeating letters and digits are allowed in the code. What is ther probability of a thief breaking into the vault if the thief has no prior knowledge of the secret code?

1 / (26⁵.10⁴)