The number of words of four letters containing equal number of vowels and consonants, repetition being allowed, is

Options

A. `105^2`
B. `210 xx 243`
C. `105 xx 243`
D. None of these

Answer

To find the number of four-letter words containing an equal number of vowels and consonants with repetition allowed, we can follow these steps:

### Step 1: Determine the Composition of the Word
Since we need a four-letter word with an equal number of vowels and consonants, we will have:
- 2 vowels
- 2 consonants

### Step 2: Choose Positions for Vowels
We need to select 2 positions out of 4 for the vowels. The number of ways to choose 2 positions from 4 is given by the combination formula \( \binom{n}{r} \):
\[ \text{Ways to choose positions for vowels} = \binom{4}{2} \]

### Step 3: Calculate \( \binom{4}{2} \)
Using the combination formula:
\[ \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

### Step 4: Determine the Number of Vowels and Consonants
Assuming we have:
- 5 vowels (A, E, I, O, U)
- 21 consonants (the remaining letters of the English alphabet)

### Step 5: Calculate the Number of Ways to Fill the Vowel and Consonant Positions
Since repetition is allowed:
- For each of the 2 vowel positions, we can choose any of the 5 vowels.
- For each of the 2 consonant positions, we can choose any of the 21 consonants.

Thus, the total number of ways to fill the vowel positions is:
\[ \text{Ways to fill vowel positions} = 5 \times 5 = 25 \]
And the total number of ways to fill the consonant positions is:
\[ \text{Ways to fill consonant positions} = 21 \times 21 = 441 \]

### Step 6: Calculate the Total Number of Words
Now, we can calculate the total number of four-letter words:
\[ \text{Total words} = \text{Ways to choose positions for vowels} \times \text{Ways to fill vowel positions} \times \text{Ways to fill consonant positions} \]
\[ \text{Total words} = 6 \times 25 \times 441 \]

### Step 7: Perform the Calculation
Calculating this step-by-step:
1. \( 6 \times 25 = 150 \)
2. \( 150 \times 441 = 66150 \)

Thus, the total number of four-letter words containing an equal number of vowels and consonants, with repetition allowed, is **66150**.