Question

1. Given a security code for door access consists of 4 alphanumeric characters, and the first character has to be a letter. Determine the proportion of security code that:
   1. is a unique code                                                                                         
   2. begin with a vowel (a, e, i, o, u)                                                                 
   3. end with odd number (1, 3, 5, 7, 9)
   4. begin with a vowel and end with odd number                 

Answer 1

a. Proportion of security code that: is a unique code

Security code :

First character has to be a letter : Number of possibilities : 26(Case not considered)

Second character : Alphanumeric : Number of possibilities : 26(letters) + 10 numeric digits = 36

Third character : Alphanumeric : Number of possibilities : 26(letters) +10 numeric digits = 36

Fourth character: Alphanumeric : Number of possibilities : 26(letters) + 10 numeric digits = 36

Total number of possibilities = 26 x 36 x 36 x 36 = 1213056

Unique code : 1

Proportion of security code that: is a unique code = 1/1213056

b. begin with a vowel (a, e, i, o, u)

First character has to be a vowel (a,e,i,o,u) : Number of possibilities :5(Case not considered)

Second character : Alphanumeric : Number of possibilities : 26(letters) + 10 numeric digits = 36

Third character : Alphanumeric : Number of possibilities : 26(letters) + 10 numeric digits = 36

Fourth character: Alphanumeric : Number of possibilities : 26(letters) + 10 numeric digits = 36

Total number of possible security codes that begin with a vowel (a, e, i, o, u) = 5*36*36*36 = 233280

Total number of possibilities = 1213056

Proportion of security code that begin with a vowel (a, e, i, o, u)= 233280/1213056 = 0.1923

c. end with odd number (1,3,5,7,9)

First character has to be a letter Number of possibilities :26(Case not considered)

Second character : Alphanumeric : Number of possibilities : 26(letters) +10 numeric digits = 36

Third character : Alphanumeric : Number of possibilities : 26(letters) +10 numeric digits = 36

Fourth character: odd number (1,3,5,7,9) : Number of possibilities : 5

Total number of possible security codes that end with odd number (1,3,5,7,9) = 26*36*36*5 = 168480

Total number of possibilities = 1213056

Proportion of security code end with odd number (1,3,5,7,9) = 168480/1213056 = 0.1389

d. begin with a vowel and end with odd number     

First character has to be a vowel (a,e,i,o,u) : Number of possibilities :5(Case not considered)

Second character : Alphanumeric : Number of possibilities : 26(letters) +10 numeric digits = 36

Third character : Alphanumeric : Number of possibilities : 26(letters) +10 numeric digits = 36

Fourth character: odd number (1,3,5,7,9) : Number of possibilities : 5

Total number of possible security codes begin with a vowel and end with odd number = 5 * 36 * 36 * 5 = 32400

Total number of possibilities = 1213056

Proportion of security code begin with a vowel and end with odd number= 32400/1213056 = 0.0267