How many different ways can the letters in the word "micro" be arranged if it always has to start with a vowel?

-- The first letter can be either one of 2 letters ('i' or 'o').  For each of those ...
-- The second letter can be any one of the remaining 4 .  For each of those ...
-- The third letter can be any one of the remaining 3 .  For each of those ...
-- The fourth letter can be any one of the remaining 2 .  For each of those ...
-- There's only 1 letter left to go into the fifth place.

Total possible arrangements = (2 x 4 x 3 x 2 x 1) = 48 possibilities.

There are two vowels in the word 'micro' i and o
so there are 2 possibilities for the first letter slot.

there are 5 slots total and the first slot has been taken so
there are 4 options for the second slot
3 options in the 3rd slot
2 options in the 4th slot
and 1 option in the 5th slot

so 2*4*3*2*1=48 possibilities