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) no.of roman greek alphabets=12

no.of roman numerals=5

since we can go for repetition of numbers or letters, the no.of secret codes that can be formed of length 10=17¹⁰

17 ways

17 ways

17 ways

17 ways

17 ways

17 ways

17 ways

17 ways

17 ways

17 ways

b) no.of secret codes with greek alphabets only is equivalent to arranging code from the 12 greek alphabets=12¹⁰

12 ways

12 ways

12 ways

12 ways

12 ways

12 ways

12 ways

12 ways

12 ways

12 ways

the probability that a randomly selected passcode contains only Greek letters and no numbers

odd-5 ways

15 ways

15 ways

15 ways

15 ways

15 ways

15 ways

15 ways

15 ways

odd-5 ways

first and last part of code must be odd

first place will be fiiled in 5 ways and so the last place

remaining odd numerals=3

so now we need to fill remaining 8 places with 3 numerals and 12 alphabets and no of ways to fill so =15⁸

required no.of ways=5*15⁸*5

probability that the first and last position of a randomly selected passcode contains odd numbers=

d) without repetition means 1st position will be filled in 17 ways: since after filling the first position there will be 16 ways left to be filled; 3rd position 15 ways........

16 ways

15 ways

14 ways

13 ways

12 ways

11 ways

10 ways

8 ways

7ways

6 ways

the probability that a randomly selected code has no repeats=e) if codes start and end with alphabet we need to fill the remaining 8 positions with the given 12 alphabets and 5 numerals since repetition is allowed

-one way

17 ways

17 ways

17 ways

17 ways

17 ways

17 ways

17 ways

17 ways

-one way

probability that it starts and ends with the letter ?=