Question

Of the 13-digit passwords formed by mixing (7 letters, from A to Z, without repeats, and 6 numbers, from 1 to 9 allowing repeats) I pick one at random. Determine Prob( my word begins with 1, ends with 9, and includes exactly 3 vowels)

Answer 1

Total letters = 26 (A,B,C,D……………X,Y,Z)

Out of these 26 letters, there are 5 vowels and 21 consonants.

Total numbers = 9 (1, 2, 3, 4, 5, 6, 7, 8, 9)

Password length = 13 digits

Total number of different passwords that can be formed with the given condition is given by sample space;

Sample space = n(S) = (26 x 25 x 24 x 23 x 22 x 21 x 20) x ( 9 x 9 x 9 x 9 x 9 x 9)

Let A =Event that the password begins with 1, ends with 9 and includes exactly 3 vowels

n(A) = 1 x 9 x 9 x 9 x 9 x 5 x 4 x 3 x 21 x 20 x 19 x 18 x 1

P(A) = n(A) / n(S)

= [ 1 x 9 x 9 x 9 x 9 x 5 x 4 x 3 x 21 x 20 x 19 x 18 x 1 ] /   [ 26 x 25 x 24 x 23 x 22 x 21 x 20 x 9 x 9 x 9 x 9 x 9 x 9 ]

= 19 / 592020

P(A) = 0.00003209