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. 128
B. 24
C. 72
D. 96

Explanation

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

### Step 1: Identify the vowels and consonants
The letters given are `a`, `b`, `c`, `d`, `e`, `f`. Among these, the vowels are `a` and `e`, and the consonants are `b`, `c`, `d`, and `f`.

### Step 2: Calculate the total number of 3-letter combinations
We can select 3 letters from the 6 available letters (a, b, c, d, e, f) and arrange them. The total number of ways to choose 3 letters from 6 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

\[ \text{Total combinations} = \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

After choosing 3 letters, we can arrange them in \( 3! \) ways.

\[ \text{Total arrangements} = 20 \times 3! = 20 \times 6 = 120 \]

### Step 3: Calculate the number of combinations without vowels
Next, we need to find the number of 3-letter combinations that can be formed using only consonants. The consonants available are `b`, `c`, `d`, and `f`, which gives us 4 consonants.

We can choose 3 consonants from these 4:

\[ \text{Combinations of consonants} = \binom{4}{3} = 4 \]

Each selection of 3 consonants can be arranged in \( 3! \) ways:

\[ \text{Arrangements of consonants} = 4 \times 3! = 4 \times 6 = 24 \]

### Step 4: Calculate the number of words with at least one vowel
To find the number of words that contain at least one vowel, we subtract the number of words that contain only consonants from the total number of arrangements:

\[ \text{Words with at least one vowel} = \text{Total arrangements} - \text{Arrangements of consonants} \]
\[ = 120 - 24 = 96 \]

### Final Answer
Thus, the number of words that can be formed from the letters `a`, `b`, `c`, `d`, `e`, `f` taken 3 together, each containing at least one vowel, is **96**.

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