SOLUTION: What is the probability on five letters that spell the word ALONE if, the first letter has to be a vowel

ALONE

First we get the numerator of the probability:

We can choose the first letter 2 ways, either an A or an O.
That's 2 ways to choose the first letter.

For each of those 2 ways to choose the first letter, there are 4 ways
to choose the 2nd letter.  That's 2×4 ways or 8 ways to choose the
first two letters.

For each of those 2×4 or 8 ways to choose the first two letters, there 
are 3 ways to choose the 3rd letter.  That's 2×4×3 ways or 24 ways to 
choose the first three letters.
 
For each of those 2×4×3 or 24 ways to choose the first three letters, there 
are 2 ways to choose the 4th letter.  That's 2×4×3×2 ways or 48 ways to 
choose the first four letters.

For each of those 2×4×3×2 or 48 ways to choose the first four letters, there 
is just 1 way to choose the 5th letter.  That's 2×4×3×2×1 ways or still
48 ways to choose all the .

Now we get the denominator.  It's like the numerator except that there are 5
ways to choose the first letter. So the denominator is 5×4×3×2×1


So the desired probability is

2×4×3×2×1
--------- =  or .4
5×4×3×2×1


Edwin