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: Understand the task.
We must form 5-letter words using exactly 3 of the 7 consonants and 2 of the 4 vowels. This is select-then-arrange.
Step 2: Choose the consonants.
Number of ways to pick 3 consonants from 7 is $\binom{7}{3} = \dfrac{7\cdot 6\cdot 5}{3\cdot 2\cdot 1} = 35$.
Step 3: Choose the vowels.
Number of ways to pick 2 vowels from 4 is $\binom{4}{2} = \dfrac{4\cdot 3}{2\cdot 1} = 6$.
Step 4: Count the selections.
Selecting the 5 letters can be done in $35 \times 6 = 210$ ways.
Step 5: Arrange the 5 letters.
Each chosen set of 5 distinct letters can be ordered in $5! = 120$ ways to make words.
Step 6: Multiply.
Total words $= 210 \times 120 = 25200$.
\[ \boxed{25200} \]