Out of 7 consonants and 4 vowels, in how many different ways can words consisting of 3 consonants and 2 vowels be formed, assuming that the letters chosen are then arranged in all possible orders?

Introduction / Context:
This question is from permutations and combinations and focuses on counting the number of words (arrangements of letters) formed by selecting consonants and vowels from given sets. It checks whether you can combine the ideas of selection and arrangement correctly.

Given Data / Assumptions:

• Number of consonants available = 7.
• Number of vowels available = 4.
• Each word must contain exactly 3 consonants and 2 vowels.
• All selected letters are arranged in all possible different orders.
• All letters are distinct and can be used at most once in each word.

Concept / Approach:
The total number of words is:

• Number of ways to choose 3 consonants out of 7 = combination C(7,3).
• Number of ways to choose 2 vowels out of 4 = combination C(4,2).
• Number of ways to arrange the selected 5 letters = 5! permutations.

Step-by-Step Solution:
Step 1: Compute C(7,3) = 7! / (3! * 4!) = (7 * 6 * 5) / (3 * 2 * 1) = 35. Step 2: Compute C(4,2) = 4! / (2! * 2!) = (4 * 3) / (2 * 1) = 6. Step 3: Compute 5! = 5 * 4 * 3 * 2 * 1 = 120. Step 4: Multiply the three results: 35 * 6 * 120. Step 5: First 35 * 6 = 210. Step 6: Then 210 * 120 = 25200. Step 7: Hence, total number of words = 25200.

Verification / Alternative check:
We can verify the reasoning logically. There are 35 possible sets of 3 consonants, and for each such set there are 6 possible sets of 2 vowels. That gives 35 * 6 = 210 distinct letter groups. For each group of 5 letters, there are 120 permutations. Therefore, 210 groups * 120 permutations per group = 25200 distinct words, which is exactly the value calculated above.

Why Other Options Are Wrong:
52000 and 24400: These are large numbers that may arise from incorrect multiplications or using 7P3 and 4P2 instead of combinations in the selection stage. 120: This is only 5!, the number of ways to arrange 5 letters, and ignores the many ways to choose the letters themselves. 8400: This might come from multiplying some but not all of the required factors correctly. None of these match the full combination and permutation reasoning.

Common Pitfalls:
A frequent error is to confuse combinations and permutations. Some students use permutations while selecting consonants and vowels, which leads to double counting. Another common issue is forgetting to include the final arrangement step after selection. Always separate the process into selection (using combinations) and arrangement (using factorials), and multiply the results in the correct order.

Final Answer:
The number of different words that can be formed is 25200.