Question 639962
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.)
<pre>

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 = 5<sup>6</sup> = 15625 ways.

Edwin</pre>