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

### Concept & Permutations with Grouping When a specific set of items must always appear together, we treat them as a single unit or "block". We then find the permutations of all units, and multiply this by the permutations of the items within the "block". The general formula for arranging $n$ distinct items is $n!$. If items are grouped into 1 unit, the total units become $(n - k + 1)$ where $k$ is the size of the group. ### Step-by-Step Solution * **Analyze the Word:** The word "JUDGE" contains 5 letters. * **Identify Vowels and Consonants:** * Vowels: U, E (2 letters) * Consonants: J, D, G (3 letters) * **Group the Vowels:** Since the vowels must be together, treat (UE) as a single unit. * **Count the Units:** We now have the consonants J, D, G and the vowel unit (UE). This gives a total of $3 + 1 = 4$ units. * **Arrange the Units:** The 4 units can be arranged in $4!$ ways. * $4! = 4 \times 3 \times 2 \times 1 = 24$ * **Arrange Items Within the Unit:** The 2 vowels (U, E) can be arranged among themselves in $2!$ ways. * $2! = 2 \times 1 = 2$ * **Calculate Total Arrangements:** Multiply the two results. * Total ways = $24 \times 2 = 48$ ### Exam Strategy & Shortcut For simple non-repeating words, immediately write down (Total Consonants + 1)! $\times$ (Total Vowels)!. Here: $4! \times 2! = 24 \times 2 = 48$. This takes less than 15 seconds. ### Common Pitfall A common mistake is forgetting to arrange the vowels within their own group. Students often calculate $4! = 24$ and stop there, resulting in an incorrect answer. Always account for internal permutations of the grouped block. ### Final Answer Therefore, the correct answer is **48**.