Timothy’s password must be five characters long. He can use zero to nine and he
cannot use the same digit more than once. How many different passwords could Timothy create?
Timothy has to make five choices. The first digit of his password can be anything from zero to nine, so how many
options does he have? The sample space there is ten, zero plus all nine of those digits, ten choices.
For the second character, something changes. We no longer have ten options to choose from because we’ve already used one of
the digits in our first character. We now only have nine options available to us to choose from.
After that, since he can’t repeat digits, he’ll have eight choices, followed by seven choices and then six choices. This progression is because order is important and he cannot repeat any digit.
To solve from here, we’ll need to remember the fundamental counting principle. If
event M has 𝑚 possible outcomes and event N has 𝑛 possible outcomes, then event M followed
by event N has 𝑚 times 𝑛 possible outcomes.
We have five events and we know how many possible outcomes for each event there
are, which means what we need to do is we need to multiply the possible outcomes from each
When you multiply ten times nine times eight times seven times six, it equals
thirty thousand two hundred and forty. These are the amount of different passwords that Timothy could create.