To open a safe five numbers must be entered on a keypad. The keypad contains numbers 0-9. The numbers have to be entered in a certain order to open the safe.
a. What are the TOTAL number of possible ways you can enter five numbers to open this safe (if there are NO restrictions on the numbers that
may be chosen)?
b. If no numbers repeat, then how many entries are possible?
c. What is the PROBABILITY that the safe's five numbers do not repeat? Write your answer as a percent rounded to the nearest tenth.

30.2%

Step-by-step explanation:

a. For each slot there are 10 possible numbers (0-9). Since there are no restrictions, the number in each slot is independent of the other. We can calculate the total number of possibilities using the product rule.

N = 10 × 10 × 10 × 10 × 10 = 10⁵

b. For the first slot, there are 10 possibilities.

For the second slot, we cannot repeat the number that we used in the first slot, so there are 9 possibilities.

For the third slot, we cannot repeat any of the previous numbers, so there are 8 possibilities.

For the fourth slot, we cannot repeat any of the previous numbers, so there are 7 possibilities.

For the fifth slot, we cannot repeat any of the previous numbers, so there are 6 possibilities.

The total number of combinations without any repetitions are:

10 × 9 × 8 × 7 × 6 = 30,240

c. The probability that the safe's five numbers do not repeat is:

P = favorable outcomes / possible outcomes

P =  30,240 / 10⁵ = 30.2%