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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.)

Sagot :

Semsee45idn

Answer:

21,168

Step-by-step 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

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