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. 48
B. 96
C. 32
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, f taken 3 at a time, with the condition that each word contains at least one vowel, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Vowels and Consonants**:
- The letters available are a, b, c, d, e, f.
- Vowels: a, e (2 vowels)
- Consonants: b, c, d, f (4 consonants)

2. **Total Combinations Without Restriction**:
- The total number of ways to choose 3 letters from 6 is given by the combination formula \( \binom{n}{r} \):
\[ \text{Total combinations} = \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

3. **Calculate Combinations with No Vowels**:
- We need to find the combinations that do not contain any vowels. This means we only use the consonants b, c, d, f.
- The number of ways to choose 3 consonants from 4 is:
\[ \text{Combinations with no vowels} = \binom{4}{3} = \frac{4!}{3!(4-3)!} = 4 \]

4. **Calculate Valid Combinations with At Least One Vowel**:
- To find the number of combinations that contain at least one vowel, we subtract the combinations with no vowels from the total combinations:
\[ \text{Valid combinations} = \text{Total combinations} - \text{Combinations with no vowels} = 20 - 4 = 16 \]

5. **Permutations of Each Combination**:
- Each selection of 3 letters can be arranged in \( 3! \) ways:
\[ \text{Arrangements for each combination} = 3! = 6 \]

6. **Total Words with At Least One Vowel**:
- Therefore, the total number of words that can be formed with at least one vowel is:
\[ \text{Total words} = \text{Valid combinations} \times \text{Arrangements} = 16 \times 6 = 96 \]

### Final Answer:
The total number of words that can be formed from the letters a, b, c, d, e, f taken 3 together, with at least one vowel, is **96**.