Question

Out of 7 consonants and 4 vowels, the number of words consisting of 3 consonants and 2 vowels are

• A. 3300
• B. 210
• C. 120
• D. 25200

Hint:

Do not stop calculating after completing the combinations phase! The value $210$ is intentionally included as option (B) to catch students who forget that a "word" requires the letters to be arranged in different positions. Always multiply your selection total by the factorial of the word length ($5! = 120$).

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given a pool of 7 consonants and 4 vowels. We need to find the total number of unique 5-letter words that can be created by selecting exactly 3 consonants and 2 vowels.

Step 2: Key Formula or Approach:
This problem requires a two-step approach: first selection, then arrangement. 1. Use combinations to select the sub-groups of letters: $\binom{n}{r} = \frac{n!}{r!(n-r)!}$.
2. Use permutations to arrange the 5 selected letters into unique words: $5!$.
$$\text{Total Words} = \binom{7}{3} \times \binom{4}{2} \times 5!$$

Step 3: Detailed Explanation:
3.

Select 3 consonants out of 7: $$\binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35$$ 4.

Select 2 vowels out of 4: $$\binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6$$ 5.

Calculate total selection groups: $$\text{Total ways to select letters} = 35 \times 6 = 210$$ 6.

Arrange the 5 chosen letters to form words: Each set of 5 letters can be rearranged in $5!$ unique ways: $$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$$ 7.

Compute total permutations: $$\text{Total unique words} = 210 \times 120 = 25200$$

Step 4: Final Answer:
The number of words that can be formed is 25200, which corresponds to option (D).