How many 7-letter words consisting of exactly 4 consonants and 3 vowels can be formed from 12 distinct consonants and 4 distinct vowels if the first 4 positions are consonants and the last 3 positions are vowels (pattern C C C C V V V)?

Introduction / Context:The original options suggest a fixed-position pattern for consonants and vowels rather than free intermixing. We will explicitly adopt the pattern C C C C V V V to avoid ambiguity and align with the given numeric scale.

Given Data / Assumptions:

• 12 distinct consonants available; choose and arrange 4 into the first four slots.
• 4 distinct vowels available; choose and arrange 3 into the last three slots.
• All letters are distinct; within each block, order matters.

Concept / Approach:Number of choices for consonant block = 12P4. Number for vowel block = 4P3. Multiply blocks because they are independent under the fixed pattern.

Step-by-Step Solution:

Verification / Alternative check:If consonants and vowels could appear in any positions (not fixed), the count would be 12C4 * 4C3 * 7!, which equals 9,979,200—far larger and inconsistent with the provided options. Hence the fixed-slot interpretation is appropriate.

Why Other Options Are Wrong:251820, 258120, 281520 are near-misses derived from arithmetic slips or using combinations instead of permutations in one of the blocks.

Common Pitfalls:Mixing up P and C; using 12C4 or 4C3 in place of 12P4 or 4P3 when the within-block order matters.

Final Answer:285120