Question

A well-known company manufactures combination locks. Their retractable cable lock works well for securing sports equipment (skis, bikes, golf equipment, etc.) as well as duffel or sports bags. A five-digit (0–9) dial combination lock secures the cable.

Suppose that you purchase one of these retractable cable locks to secure your bike to a bike rack at the library. How many five-digit combination lock codes are possible for your lock if you cannot repeat digits? Is this problem a combination or a permutation problem? Why?

Answer 1

Part 1: How many five-digit combination lock codes are possible for your lock?

The first digit in the lock can be chosen in 10 ways (any of the digits from 0 to 9).

The second digit in the lock can be chosen in 9 ways (any of the digits from 0 to 9 except the number in first digit, since digits cannot be repeated).

The third digit in the lock can be chosen in 8 ways (any of the digits from 0 to 9 except the numbers in first and second digits, since digits cannot be repeated).

The fourth digit in the lock can be chosen in 7 ways (any of the digits from 0 to 9 except the numbers in first, second and third digits, since digits cannot be repeated).

The fifth digit in the lock can be chosen in 6 ways (any of the digits from 0 to 9 except the numbers in first, second, third and fourth digits, since digits cannot be repeated).

Thus, the number of five-digit combination lock codes which are possible for our lock = 10*9*8*7*6 = 30240 [Answer]

Part 2: Combination or permutation

This problem is a Permutation problem since the ordering of the digits is of relevance here. A combination of 12345 is not the same as 54321 as the ordering is important. Had the order of the digits not been of relevance, it would have been a combination problem.

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