In how many different ways can the letters of the word SOFTWARE be arranged in such a way that the vowels always come together?

### Concept & Permutations with Grouping When grouping elements, the grouped elements are treated as a single structural unit. The fundamental principle of counting implies we multiply the permutations of the macro-structure by the permutations of the micro-structure (the grouped elements). ### Step-by-Step Solution * **Analyze the Word:** The word "SOFTWARE" contains 8 letters. * **Identify Vowels and Consonants:** * Vowels: O, A, E (3 letters) * Consonants: S, F, T, W, R (5 letters) * **Group the Vowels:** Treat the 3 vowels (O, A, E) as a single unit. * **Count the Units:** We have 5 consonants + 1 vowel unit = 6 total units. * **Arrange the Units:** The 6 units can be arranged in $6!$ ways. * $6! = 720$ * **Arrange Items Within the Unit:** The 3 vowels can be internally arranged in $3!$ ways. * $3! = 6$ * **Calculate Total Arrangements:** * Total ways = $720 \times 6 = 4320$ ### Exam Strategy & Shortcut Memorize basic factorials up to $7!$ to save time. Knowing $6! = 720$ and $3! = 6$ lets you immediately calculate $720 \times 6 = 4320$ without writing out expanded multiplication steps. ### Common Pitfall A common error is grouping the vowels but counting the remaining consonants incorrectly. Always double-check that your total letter count matches the sum of vowels and consonants. ### Final Answer Therefore, the correct answer is **4320**.