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

### Concept & Permutations with Repetition When arranging a set of items where some items are identical, the standard factorial is divided by the factorials of the counts of each identical item to remove duplicate permutations. Formula for arrangements of $n$ items with $p$ identical items of one kind: $\frac{n!}{p!}$ ### Step-by-Step Solution * **Analyze the Word:** The word "BANKING" contains 7 letters. * **Identify Vowels and Consonants:** * Vowels: A, I (2 letters) * Consonants: B, N, K, N, G (5 letters) * Notice that the letter 'N' repeats 2 times. * **Group the Vowels:** Treat (A, I) as 1 unit. * **Count the Units:** 5 consonants + 1 vowel unit = 6 units. * **Arrange the Units:** Since 'N' is repeated twice among the consonants, the arrangement of these 6 units requires dividing by $2!$. * Arrangements of units = $\frac{6!}{2!} = \frac{720}{2} = 360$ * **Arrange Items Within the Unit:** The 2 vowels (A, I) arrange in $2!$ ways. * $2! = 2$ * **Calculate Total Arrangements:** * Total ways = $360 \times 2 = 720$ ### Exam Strategy & Shortcut Whenever there are repeating letters, handle the repetitions immediately in the respective block (either the main unit arrangement or the internal vowel arrangement). Don't wait until the final multiplication step to divide, as numbers get unnecessarily large. ### Common Pitfall A major trap is forgetting to divide by the factorial of repeating letters, which would lead to an inflated answer (in this case, 1440). Always scan the word for duplicate letters before calculating. ### Final Answer Therefore, the correct answer is **720**.