Question

Out of 5 consonants and 4 vowels, how many words of 3 consonants and 3 vowels can be made?

• A. 40
• B. 80
• C. 20
• D. 240

Hint:

For such combinatorial problems, break down the task into choosing and arranging items, and apply the combination and permutation formulas.

Answer 1

The Correct Option is B

Step 1: Understand the problem.
We need to form words using 3 consonants and 3 vowels, chosen from 5 consonants and 4 vowels.

Step 2: Calculate the number of ways to choose 3 consonants and 3 vowels.
- The number of ways to choose 3 consonants from 5: \[ \binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10 \] - The number of ways to choose 3 vowels from 4: \[ \binom{4}{3} = \frac{4 \times 3 \times 2}{3 \times 2 \times 1} = 4 \]

Step 3: Multiply the results and consider arrangements.
Now, the 6 letters (3 consonants and 3 vowels) can be arranged in: \[ \frac{6!}{3!3!} = \frac{720}{6 \times 6} = 20 \] Thus, the total number of words that can be made is: \[ 10 \times 4 \times 20 = 80 \] Therefore, the correct answer is 2. 80.