How many 4-digit license plates can be made by using
the digits 0, 2, 4, 6, and 8 if repetitions are allowed? if
repetitions are not allowed?
To find the number of 4-digit license plates that can be made using the digits 0, 2, 4, 6, and 8, we can consider both cases: repetitions allowed and repetitions not allowed.
1. Repetitions allowed:
In this case, we can choose any digit from the given set (0, 2, 4, 6, 8) for each of the four positions independently. Since there are five choices for each position, the total number of possible license plates is calculated as the product of these choices:
5 choices × 5 choices × 5 choices × 5 choices = 5^4 = 625 possible license plates.
2. Repetitions not allowed:
In this case, we can choose a digit for the first position in five ways (as there are five digits available). However, for the second position, we can choose from the remaining four digits because we cannot repeat the digit used in the first position. Similarly, for the third position, we will have three choices, and for the fourth position, there will be two choices. Therefore, the total number of possible license plates without repetition can be calculated as follows:
5 choices × 4 choices × 3 choices × 2 choices = 5 × 4 × 3 × 2 = 120 possible license plates.
So, with repetitions allowed, there can be 625 possible license plates, and without repetitions allowed, there can be 120 possible license plates.