Question

2. 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
a) 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
b) 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

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

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

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