5. In how many ways can the letters of olympic be arranged if
a) there are no restrictions?
b) consonants and vowels (o, i, and y) alternate?
c) all vowels are in the middle of each arrangement?

Step 1 ：Calculate the total number of arrangements with no restrictions: \(7! = 5040\)

Step 2 ：Calculate the number of arrangements for consonants: \(4! = 24\)

Step 3 ：Calculate the number of arrangements for vowels: \(3! = 6\)

Step 4 ：Calculate the number of alternating arrangements: \(2 \times 24 \times 6 = 288\)

Step 5 ：Calculate the number of arrangements with vowels in the middle: \(24 \times 6 = 144\)

Step 6 ：\(\boxed{\text{Final Answer:}}\)

Step 7 ：a) \(\boxed{5040}\) ways with no restrictions

Step 8 ：b) \(\boxed{288}\) ways with consonants and vowels alternating

Step 9 ：c) \(\boxed{144}\) ways with all vowels in the middle