The number of words of four letters consisting of equal number of vowels and consonants (of English Language) with repetition permitted is

Options

A. 51030
B. 50030
C. 63050
D. 66150

Answer

To solve the problem of finding the number of four-letter words consisting of an equal number of vowels and consonants, we can follow these steps:

### Step 1: Understand the Requirements
We need to create a four-letter word with an equal number of vowels and consonants. This means we will have 2 vowels and 2 consonants.

### Step 2: Identify the Number of Vowels and Consonants
In the English language:
- The number of vowels (A, E, I, O, U) = 5
- The number of consonants = 26 (total letters) - 5 (vowels) = 21

### Step 3: Choose Positions for Vowels
We need to choose 2 positions out of the 4 available for the vowels. The number of ways to choose 2 positions from 4 is given by the combination formula \( \binom{n}{r} \):
\[ \text{Number of ways to choose positions} = \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6 \]

### Step 4: Fill the Chosen Positions with Vowels
Since repetition is allowed, each of the 2 vowel positions can be filled with any of the 5 vowels. Therefore, the number of ways to fill the vowel positions is:
\[ 5 \times 5 = 25 \]

### Step 5: Fill the Remaining Positions with Consonants
The remaining 2 positions will be filled with consonants. Since repetition is allowed, each of the 2 consonant positions can be filled with any of the 21 consonants. Therefore, the number of ways to fill the consonant positions is:
\[ 21 \times 21 = 441 \]

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

### Step 7: Perform the Calculation
Now we calculate:
\[ 6 \times 25 = 150 \]
\[ 150 \times 441 = 66150 \]

### Final Answer
Thus, the total number of four-letter words consisting of an equal number of vowels and consonants is **66150**.

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