Question

The access code to a house's security system consists of eight digits. How many different codes are
available if each digit can be repeated?

Answer 1

Answer on Question #46748 – Math – Combinatorics | Number Theory

PROBLEM:

The access code to a house's security system consists of eight digits. How many different codes are available if each digit can be repeated?

SOLUTION:

Consider one of eight positions separately from all others. There are 10 possible ways to choose a digit to place on this position since there are ten digits from 0 to 9, and this choice is independent of all other positions since each digit can be repeated and there is no restriction on how to choose digits.

So, we have 8 positions and each one is filled by a digit in 10 ways independently from each other, therefore the total number of possible codes equals to ways it can be done for each independent position multiplied altogether. This is $10 \times 10 \times \ldots \times 10 = 10^{8}$ (100 million).

Answer: $10^{8}$ different codes.

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