1. **Problem statement:** Find the number of ways to arrange the letters of the word INFORMATION such that all vowels are together. 2. **Identify vowels and consonants:** The word INFORMATION has vowels I, O, A, I, O (5 vowels) and consonants N, F, R, M, T, N (6 consonants). 3. **Treat all vowels as a single unit:** Since all vowels must be together, consider the block of vowels as one unit. Along with the 6 consonants, we have 7 units total: (Vowel block) + N + F + R + M + T + N. 4. **Count arrangements of these 7 units:** The consonant N appears twice, so the number of ways to arrange these 7 units is $$\frac{7!}{2!}$$. 5. **Count arrangements of vowels inside the block:** The vowels are I, O, A, I, O with I and O repeated twice each. The number of ways to arrange these vowels is $$\frac{5!}{2!2!}$$. 6. **Calculate total arrangements:** Multiply the arrangements of the 7 units by the arrangements of vowels inside the block: $$\frac{7!}{2!} \times \frac{5!}{2!2!} = \frac{5040}{2} \times \frac{120}{4} = 2520 \times 30 = 75600$$. **Final answer:** There are 75600 ways to arrange the letters of INFORMATION such that all vowels are together.