1. **Problem statement:** Find the number of ways to arrange the letters of "olympic" under two conditions: 2. **Part (a): No restrictions** - The word "olympic" has 7 distinct letters: o, l, y, m, p, i, c. - The number of ways to arrange $n$ distinct letters is $n!$. - Here, $n=7$, so the total arrangements are: $$7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040$$ 3. **Part (b): Consonants and vowels alternate** - Vowels in "olympic" are o, i, y (3 vowels). - Consonants are l, m, p, c (4 consonants). - Since there are more consonants, the arrangement must start and end with consonants to alternate properly. - Positions: C V C V C V C (7 positions) 4. **Number of ways to arrange consonants:** - There are 4 consonants to place in 4 consonant positions. - Number of ways: $4! = 24$ 5. **Number of ways to arrange vowels:** - There are 3 vowels to place in 3 vowel positions. - Number of ways: $3! = 6$ 6. **Total arrangements with alternating consonants and vowels:** - Multiply consonant and vowel arrangements: $$4! \times 3! = 24 \times 6 = 144$$ **Final answers:** - (a) $5040$ - (b) $144$