In how many different ways can the letters of the word LEADING be arranged such that all the vowels in the word always come together as one block?

Introduction / Context:
This problem checks your understanding of permutations under a grouping condition, where certain letters (the vowels) must always appear together. Instead of treating all letters as separate, we temporarily treat the group of vowels as one combined unit. This idea is very common in permutation and combination questions and helps to simplify complex arrangement problems.

Given Data / Assumptions:
The word is LEADING. Letters in LEADING are L, E, A, D, I, N, G. Vowels: E, A, I. Consonants: L, D, N, G. All seven letters are distinct, so there are no repetitions. Vowels must always stay together as a single contiguous block.

Concept / Approach:
To ensure that vowels stay together, we group E, A and I into one block. This block acts like a single super letter. Together with the consonants L, D, N and G, we then count the permutations of these objects. Finally, we multiply this by the number of internal permutations of the vowels within their block. This multiplication rule arises because for each arrangement of the blocks, there are several internal arrangements of vowels.

Step-by-Step Solution:
Step 1: Treat E, A and I as one block which we can call V. Step 2: Now we have the objects: V, L, D, N, G. That is 5 distinct objects to arrange. Step 3: The number of ways to arrange 5 distinct objects in a line is 5! = 120. Step 4: Inside the vowel block V, the three vowels E, A and I can be arranged in 3! = 6 ways. Step 5: For each arrangement of the 5 objects, there are 6 internal arrangements of the vowels. Step 6: Total required arrangements = 5! * 3! = 120 * 6 = 720.

Verification / Alternative check:
We can quickly confirm the logic by checking extremes. If there were no restriction, the total permutations of LEADING would be 7! = 5040. Requiring vowels to be together naturally reduces this count. The calculated answer 720 is significantly less than 5040, which makes sense. Also, another mental check is to think of the positions of the vowel block: as we have 5 objects including the block, the block can appear in any of 5 positions relative to the consonants, and for each such placement the three vowels can permute among themselves in 6 ways. It is consistent with the 5! * 3! calculation.

Why Other Options Are Wrong:
Values like 520, 700 and 750 do not follow the factorial based reasoning arising from the grouping method. They are often the result of guessing or partial multiplication mistakes. For example, mistakenly using 4! instead of 5!, or forgetting to account for the internal arrangement of vowels, would produce wrong totals. Only 720 correctly represents the number of arrangements obtained by multiplying 5! and 3! together.

Common Pitfalls:
A very common mistake is to forget that the vowels themselves can be arranged in different orders inside the block, leading to undercounting. Another mistake is to count the arrangement of the 5 objects incorrectly, or to accidentally include arrangements where vowels are not adjacent. Some students treat vowels as fixed in a particular order, which again reduces the total count incorrectly. Always remember to treat the vowel block as one object first, arrange all objects, and then expand the block internally.

Final Answer:
The number of different arrangements where all vowels in LEADING come together is 720.