Number of arrangements of the letters of the word BENGALI when all three vowels always appear together

Introduction / Context:
This question involves permutations of the letters of a word with the condition that all vowels must be grouped together. Such problems test your ability to use the block method and count permutations when there are both consonants and vowels.

Given Data / Assumptions:

• Word: BENGALI.
• Letters: B, E, N, G, A, L, I.
• All seven letters are distinct.
• Vowels: E, A, I.
• Consonants: B, N, G, L.
• The three vowels must always appear together as one group in every arrangement.

Concept / Approach:
We treat the group of vowels as a single block so that they stay together. Then we arrange this block along with the consonants. Finally, we count the internal arrangements of the vowels inside their block and multiply the two counts.

Step-by-Step Solution:
Step 1: Group the vowels E, A, I into a single block V.Step 2: The consonants are B, N, G, L, which are four distinct letters.Step 3: Now we have five items to arrange: V, B, N, G, L.Step 4: Number of permutations of these five distinct items = 5! = 120.Step 5: Inside the vowel block V, the letters are E, A, I, all distinct.Step 6: These three vowels can be arranged among themselves in 3! = 6 ways.Step 7: Total arrangements with vowels always together = 5! * 3! = 120 * 6 = 720.

Verification / Alternative check:
If you wish, you can think of first arranging the four consonants in 4! ways and then inserting the vowel block V in any of the five possible slots around these consonants. That again gives 4! * 5 * 3! = 24 * 5 * 6 = 720, which matches the earlier calculation, because 5! = 4! * 5.

Why Other Options Are Wrong:

• 120 counts only the arrangements of the five items V, B, N, G, L and ignores internal vowel permutations.
• 640 and 540 do not align with any standard factorial or block count, so they are incorrect.

Common Pitfalls:
Students sometimes misread the condition and try to count arrangements where no two vowels are together instead of all vowels together. Others forget to account for the internal arrangements of vowels and stop at 5!. When dealing with block problems, always consider both the outer arrangement of blocks and the inner arrangement within each block.

Final Answer:
The number of arrangements of BENGALI with all vowels together is 720, so the correct answer is 720.