c) If repetitions are allowed, as in part (b), how many of the plates have only vowels (A,E, I, O, U) and even digits? (0 is an even integer.)
There are 5 vowels {A,E,I,O,U} and also 5 even digits, {0,2,4,6,8}
Let's look an a sample license plate
________________
| |
| E U 4 0 6 4 |
|________________|
1. There are 5 ways to choose a letter to go where the E is. That's 5 ways to
place just the first letter
2. For each of those 5 ways to just put a vowel where the E is, there are 5
ways to put a vowel where the U is. So that's 5×5 or 25 ways to put
vowels just where the E and U are.
3. For each of those 5×5 or 25 ways to put vowels where the E and U are, there
are 5 ways to put an even digit where the first 4 is. So that's 5×5×5 or 25×5
or 125 ways to put vowels where the E and U are and an even digit where the
first 4 is.
3. For each of those 5×5×5 or 125 ways to put vowels where the E and U are and
an even digit where the first 4 is, there are 5 ways to put an even digit where
the 0 is. So that's 5×5×5×5 or 125×5 or 625 ways to put vowels where the E and
U and even digits where the first 4 and the 0 are.
4. For each of those 5×5×5×5 or 625 ways to put vowels where the E and U are
and even digits where the first 4 and the 0 are, there are 5 ways to put an
even digit where the 6 is. So that's 5×5×5×5×5 or 625×5 or 3125 ways to put
vowels where the E and U are and even digits where the first 4, the 0 and the 6
are.
5. For each of those 5×5×5×5×5 or 3125 ways to put vowels where the E and U are
and even digits where the first 4, the 0, and the 6 are, there are 5 ways to
put an even digit where the second 4 is. So that's 5×5×5×5×5×5 or 3125×5 or
15625 ways to put vowels where the E and U are and even digits where the first
4, the 0, the 6 and the second 4 are.
Answer: 5×5×5×5×5×5 = 56 = 15625 ways.
Edwin