Quincy's PIN must be a 4-digit number. (a) Quincy asks the bank representative if digits can be used more than once. The
representative says, "Yes, allowing for repetition reduces the likelihood of someone
guessing your password." Explain why this is true and calculate the probability of
someone guessing a PIN in one try under each scenario.

Combinations allow 10,000, permutations allow 5,040.

Step-by-step explanation:

If we allow for repetition, we can have 10^4 possible combinations, if we can start numbers with 0. This is a base 10 system with 4 different objects.

If we don't allow for repetition, we will have 10 possible objects and a subset of 4. Every time we use a digit, we cannot reuse it, so we first have 10 possible digits for the 1st digit.

We have 10*9*8*7 possible combinations if we were to fill in the rest.

0XXX

Since we chose 0, we have 9 other digits remaining (9*8*7 possible permutations)

01XX

8 other digits. (8*7 possible permutations in this system)

018X

7 digits are possible now. (7 possible permutations if we pick 1 more digit)

In conclusion, we would have 10*9*8*7 (10!/6! or 5040) possible permutations in this system. Compared to using combinations where order and reuse is allowed, we have 10^4 possible combinations, or 10000. Combinations win.