In how many ways can the letters of the word "COMPUTER" be arranged, and what is the probability that all the vowels in these arrangements always appear together?

Word: COMPUTER Total letters = 8 (all different) Total number of arrangements: 8! = 40320 Vowels and consonants: Vowels: O, U, E (3 vowels) Consonants: C, M, P, T, R (5 consonants) Treat all vowels as one block [OUE]. Then we have: [OUE], C, M, P, T, R → total 6 items. Number of ways to arrange these 6 items: 6! = 720 Number of ways to arrange the 3 vowels within the block: 3! = 6 So, number of arrangements with all vowels together: 6! × 3! = 720 × 6 = 4320 Probability that all vowels are together: Probability = (favourable arrangements) / (total arrangements) = 4320 / 40320 = 3 / 28