Question

Jerome is creating a secret passcode for his vault. The vault uses some of the Greek alphabet –possible choices are (?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?, ?}, and can also use odd digits (1, 3, 5, 7, 9). The code will be of length 10 and selected randomly from the possible Greek letters or Roman Numerals mentioned above.

a) How many possible secret passcodes can be formed of length 10?

b) What is the probability that a randomly selected passcode contains only Greek letters and no numbers?

c) What is the probability that the first and last position of a randomly selected passcode contains odd numbers?

d) What is the probability that a randomly selected code has no repeats?

e) Given that a randomly selected passcode contains only Greek letters, what is the probability that it starts and ends with the letter ??

Answer 1

(a)

Number of possible secret passcodes can be formed of length 10 = 17¹⁰ = 2,015,993,900,449

So,

Answer is:

2,015,993,900,449

(b)

Number of possible passcode containing only Greek letters and no numbers = 12¹⁰ = 61,917,364,224

So,

P( a randomly selected passcode contains only Greek letters and no numbers) = 61,917,364,224/ 2,015,993,900,449

                                                                                                                  = 0.0307

So,

Answer is:

0.0307

(c)

First & Last letter can be selected in 5 ways each.

Second to 9th letter can be selected in 17 ways.

So,

Number of possible passcode where the first and last position of a randomly selected passcode contains odd numbers = 5² X 17⁸ = 25 X 6,975,757,441 = 174,393,936,025

So,

P( the first and last position of a randomly selected passcode contains odd numbers) = 174,393,936,025/ 2,015,993,900,449

                                                                                                                               = 0.0865

So,

Answer is:

0.0865

(d)

Number of possible passcode containing no repeats = 17 X 16 X 15 X 14 X 13 X 12 X 11 X 10 X 9 X 8 = 70,752,902,400

So,

P( a randomly selected code has no repeats ) = 70,752,902,400/ 2,015,993,900,449

                                                                   = 0.0350

So,

Answer is:

0.0350

(e)
P( it starts and ends with the letter ? / contains only Greek letters) = P(it starts and ends with the letter ? AND contains only Greek letters) / P(contains only Greek letters)

= 12⁸ /12¹⁰

= 1/12²

= 0.0069

So,

Answer is:

0.0069