Question

A password has to use the following format: LDDDDLLL, where L can be any of the upper case letters in the alphabet, and D can be any digit including 0. None of the digits or letters can repeat. 

1. What is the probability that a random password uses only vowels (A,E,I,O,U) and odd numbers (1,3,5,7,9)? Show work but do not evaluate.
2. What is the probability that the password spells the word “MATH?” Show work but do not evaluate. 

Answer 1

a) Since there are 26 letters in the English alphabet and we need to put 5 letters in 4 positions, and also choose 5 out of 10 numbers in 4 positions, excluding each time the symbol is used, the probability will be equal to

$(\frac{5}{26}\cdot\frac{4}{25}\cdot\frac{3}{24}\cdot\frac{2}{23})\cdot(\frac{5}{10}\cdot\frac{4}{9}\cdot\frac{3}{8}\cdot\frac{2}{7})$

or

$\frac{C(5,4)}{C(26,4)}\cdot\frac{C(5,4)}{C(10,4)}=\frac{5!4!22!}{4!26!}\cdot\frac{5!4!6!}{4!10!}=\frac{5\cdot4\cdot3\cdot2}{26\cdot25\cdot24\cdot23}\cdot\frac{5\cdot4\cdot3\cdot2}{10\cdot9\cdot8\cdot7}$

b) Each of the 4 positions must contain a specific letter of the alphabet , and the positions of the numbers can have any correct combinations, the probability will be equal to

$\frac{1}{26}\cdot\frac{1}{25}\cdot\frac{1}{24}\cdot\frac{1}{23}$