Porisma

Worked solution

Count the settings of a four-wheel lock

Do you know the total number of combinations possible on a combination lock which contains 4 wheels each numbered 0–9?

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Schematic of the source four-wheel lock: each labelled wheel can display a digit from zero to nine. Every wheel has ten choices, repetition is allowed, and leading zero is allowed.
Schematic of the source four-wheel lock: each labelled wheel can display a digit from zero to nine. Every wheel has ten choices, repetition is allowed, and leading zero is allowed.
Schematic of the source four-wheel lock: each labelled wheel can display a digit from zero to nine. Every wheel has ten choices, repetition is allowed, and leading zero is allowed.

CJC · Lecture · Opening lock question (unnumbered)

Define the outcome as four labelled wheel positions

Each wheel can show one of the ten digits 0–9. The wheels are different positions, digits may repeat and a leading 0 is allowed. The word combination here means a lock setting; swapping different wheel digits changes the setting.

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Multiply the choices at all four positions

There are 10×10×10×10=104=10,00010\times10\times10\times10=10^4=10{,}000 settings, from 0000 through 9999. This supplies the count invited by the opening question; the source itself leaves it as a motivating question.

How did this part go?

Common mistakes

A lock code is an ordered string, even though everyday language calls it a combination. Do not exclude a leading zero or disallow repeated digits.

Read the hint