How many distinct 5-letter arrangements can be formed from the letters of the word TRIGONAL if each arrangement uses exactly two of the vowels (A, I, O) and exactly three of the consonants, with no letter repeated?

Introduction / Context:
This problem tests core ideas from permutations and combinations. The word TRIGONAL has several distinct letters, including both vowels and consonants. We are asked to build 5-letter arrangements that obey two simultaneous constraints: exactly two vowels must be used and exactly three consonants must be used, with each letter used at most once. Such questions are common in aptitude exams because they combine selection (combinations) with arrangement (permutations), and they require careful counting of how many choices are possible at each stage.

Given Data / Assumptions:

• The word TRIGONAL has 8 distinct letters: T, R, I, G, O, N, A, L.
• Vowels in the word: A, I, O (3 vowels in total).
• Consonants in the word: T, R, G, N, L (5 consonants in total).
• We must form 5-letter arrangements.
• Exactly 2 vowels and exactly 3 consonants must be used.
• No letter may be repeated within an arrangement.

Concept / Approach:
The solution proceeds in two stages. First, we choose which actual letters will be included in each 5-letter arrangement. Second, we count how many different ways these chosen letters can be permuted. Choosing letters is a combinations problem because we are selecting subsets of vowels and consonants. Arranging the chosen letters is a permutations problem because order matters in forming different 5-letter words. The total number of valid arrangements is the product of the number of ways to choose the letters and the number of ways to arrange them.

Step-by-Step Solution:
Step 1: Choose 2 vowels out of the 3 vowels A, I, O. Number of ways to choose the vowels = C(3, 2). C(3, 2) = 3. Step 2: Choose 3 consonants out of the 5 consonants T, R, G, N, L. Number of ways to choose consonants = C(5, 3). C(5, 3) = 10. Step 3: For every chosen set of 5 letters (2 vowels + 3 consonants), arrange them in all possible orders. Number of arrangements of 5 distinct letters = 5! = 120. Step 4: Multiply all stages to get the total number of valid 5-letter arrangements. Total arrangements = C(3, 2) * C(5, 3) * 5!. Total arrangements = 3 * 10 * 120 = 3600.

Verification / Alternative check:
Another way to verify is to list the counting logic verbally: there are 3 ways to choose which 2 vowels appear, 10 ways to choose which 3 consonants appear, and for each specific choice of 5 letters, there are 120 different orders. Because all letters are distinct, no overcounting occurs. The multiplication principle therefore applies correctly, and the final number 3600 is consistent with standard combinatorial reasoning.

Why Other Options Are Wrong:
Option 6300 overcounts and does not correspond to any correct combination and permutation breakdown. Option 2400 is too small; it would arise only if either the number of vowel choices or consonant choices was miscounted. Option 7200 doubles the correct answer, which usually comes from mistakenly allowing 6 letters instead of 5 or from counting some step twice. Option 1800 is exactly half of the correct total, which could result from forgetting the contribution of either vowel choices or consonant choices.

Common Pitfalls:
A common mistake is to choose the total 5 letters directly from all 8 letters and then try to adjust for the vowel and consonant constraints, which is more complicated and error prone. Another typical error is to forget that the letters are distinct, which might lead to wrong use of factorials or division by extra symmetry factors. Some learners also accidentally use permutations instead of combinations when selecting vowels and consonants, or they forget to multiply by 5! at the end. Carefully separating the selection stage from the arrangement stage avoids these issues.

Final Answer:
Thus, the total number of distinct 5-letter arrangements of TRIGONAL that contain exactly two vowels and three consonants is 3600.