To cancel an electronic alarm, a 5-digit code number must be entered into the code box. Assuming that digits may be repeated, how many codes are possible?
There are ten digits: 0, 1, 2,...8, 9
Think of this question from perspective of the person who sets the code.
When he picks the first of the 5 digits, how many choices does he have? Exactly 10 choices!
When he picks the second of his 5 digits, he can again choose from 10 options.
For digits 3, 4 and 5 he also has 10 choices.
So he has 10*10*10*10*10 choices or 10^5.
To contract: if repetition is not allowed, there are
10*9*8*7*6 choices.
For the first digit he can choose any number 0-9.
For the second digit he can choose any number 0-9 EXCEPT the number he chose for the first digit, so he has 9 options.
And so on.