The number of words which can be formed out of the letters `a`, `b`, `c`, `d`, `e` `f` taken 3 together, each word containing one vowel at least is

Options

A. 72
B. 48
C. 96
D. none of these

Answer

To solve the problem of finding the number of words that can be formed from the letters `a`, `b`, `c`, `d`, `e`, and `f` taken 3 at a time, with the condition that each word must contain at least one vowel, we can break it down into steps:

### Step 1: Identify the letters and categorize them
We have the letters: `a`, `b`, `c`, `d`, `e`, `f`.
- Vowels: `a`, `e` (2 vowels)
- Consonants: `b`, `c`, `d`, `f` (4 consonants)

### Step 2: Determine the cases based on the number of vowels
Since we need at least one vowel in each word, we can have two cases:
1. Case 1: 1 vowel and 2 consonants
2. Case 2: 2 vowels and 1 consonant

### Step 3: Calculate the number of arrangements for Case 1 (1 vowel and 2 consonants)
- **Choosing 1 vowel from 2 vowels**: This can be done in \( \binom{2}{1} = 2 \) ways.
- **Choosing 2 consonants from 4 consonants**: This can be done in \( \binom{4}{2} = 6 \) ways.
- **Arranging the 3 letters (1 vowel + 2 consonants)**: The arrangement can be done in \( 3! = 6 \) ways.

So, the total for Case 1 is:
\[ \text{Total for Case 1} = \binom{2}{1} \times \binom{4}{2} \times 3! = 2 \times 6 \times 6 = 72 \]

### Step 4: Calculate the number of arrangements for Case 2 (2 vowels and 1 consonant)
- **Choosing 2 vowels from 2 vowels**: This can be done in \( \binom{2}{2} = 1 \) way.
- **Choosing 1 consonant from 4 consonants**: This can be done in \( \binom{4}{1} = 4 \) ways.
- **Arranging the 3 letters (2 vowels + 1 consonant)**: The arrangement can be done in \( 3! = 6 \) ways.

So, the total for Case 2 is:
\[ \text{Total for Case 2} = \binom{2}{2} \times \binom{4}{1} \times 3! = 1 \times 4 \times 6 = 24 \]

### Step 5: Combine the results from both cases
Now, we add the totals from both cases to get the final answer:
\[ \text{Total number of words} = \text{Total for Case 1} + \text{Total for Case 2} = 72 + 24 = 96 \]

### Final Answer
The total number of words that can be formed is **96**.

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