Question

To use an automated teller machine at a certain​ bank, you must enter a 55​-digit ​code, using the digits 1 dash 91–9. how many 55​-digit codes are possible if repetition of digits is​ permitted?

Answer 1

There would be 15,120 different possibilities.

I assume you meant 5 digit codes and not 55 (just a typo).

There are 9 choices for the first number, then 8, then 7, then 6, then 5.
If we multiply them all together, we will have the total number of possibilities.

9 x 8 x 7 x 6 x 5 = 15120

Answer 2

There are 59,049 possibilities.

There are 9 digits possible for each of the 5 digits:
9^5 = 59049