Question

How many 5 number zip codes can be created if the first number cannot be one, three, or five, and each of the numbers must be different? (Hint: It may be helpful to list out the digits first.)

Answer 1

21,168

Explanation:

Given:

• Available digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
• Restrictions: first number can only be 0, 2, 4, 6, 7, 8, 9 (7 digits)
• Each of the 5 numbers must be different

Therefore:

• 1st number: 0, 2, 4, 6, 7, 8, 9 (7 digits)
• 2nd number: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
• 3rd number: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
• 4th number: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)
• 5th number: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (10 digits)

As each of the numbers must be different, the choices are:

• 1st number: 7 choices
• 2nd number: 10 - 1 = 9 choices
  (since we can't repeat the 1st number)
• 3rd number: 10 - 2 = 8 choices
  (since we can't repeat the 1st and 2nd numbers)
• 4th number: 10 - 3 = 7 choices
  (since we can't repeat the 1st, 2nd & 3rd numbers)
• 5th number: 10 - 4 = 6 choices
  (since we can't repeat the 1st, 2nd, 3rd & 4th numbers)

Therefore, the total number of combinations is:

7 × 9 × 8 × 7 × 6 = 21,168