SOLUTION: in how many ways cold the letters in the word "storage" be arranged if the vowels must remain in a)even positions? b)odd positions? c)even or odd positions?

There are three vowels {o,a,e} and 4 consonants {s,t,r,g}

These would would have the form

CVCVCVC 

where C represents a consonant and V a vowel.

Answer 4!3! = 144 ways.

There are 4 odd positions, 1st, 3rd, 5th, and 7th.

Thats 4C3 = 4 ways to choose the positions for
the vowels: VCVCVCC, VCVCCCV, VCCCVCV, CCVCVCV

For each of these 4 ways there are 4!3! = 144 ways to arrange them.

Answer: (4C3)(4!3!) = 4(144) = 576.

Edwin