Out of 10 consonants and 12 vowels, how many groups of 5 consonants and 5 vowels can be formed?

We need to select 5 consonants from 10 and 5 vowels from 12. Since order doesn't matter in forming groups, we use combinations. The total number of groups is C(10,5) × C(12,5) = 252 × 792 = 199584. Wait—let me recalculate: C(10,5) = 10!/(5!×5!) = 252 and C(12,5) = 12!/(5!×7!) = 792, so 252 × 792 = 199,584, which matches option B. However, option A (185252) suggests a different calculation. Re-verifying: 252 × 792 = 199,584 definitively. The correct answer should be B, not A—there may be an error in the provided answer key, but based on standard combinatorial principles, B is mathematically correct.

Step-by-step Derivation:
Step 1: Identify the problem structure.

• We need to select 5 consonants from 10 available consonants: C(10,5)
• We need to select 5 vowels from 12 available vowels: C(12,5)
• Total groups = C(10,5) × C(12,5)

Step 2: Calculate C(10,5).
C(10,5) = 10! / (5! × 5!)
= (10 × 9 × 8 × 7 × 6) / (5 × 4 × 3 × 2 × 1)
= 30240 / 120
= 252

Step 3: Calculate C(12,5).
C(12,5) = 12! / (5! × 7!)
= (12 × 11 × 10 × 9 × 8) / (5 × 4 × 3 × 2 × 1)
= 95040 / 120
= 792

Step 4: Multiply the results.
Total = 252 × 792 = 199,584

Conclusion: The mathematically correct answer is 199,584 (Option B), not 185,252 (Option A). If the answer key specifies A, there may be a transcription error or alternative interpretation in the assessment system.