Write the total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants.

Options

A. 60
B. 120
C. 7200
D. 720

Explanation

To solve the problem of finding the total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants, we can follow these steps:

### Step 1: Identify the total number of vowels and consonants
We have:
- Total vowels (V) = 4
- Total consonants (C) = 5

### Step 2: Choose the vowels
We need to choose 2 vowels from the 4 available vowels. The number of ways to choose 2 vowels from 4 can be calculated using the combination formula:

\[ \text{Number of ways to choose 2 vowels} = \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4!}{2! \cdot 2!} \]

Calculating this gives:

\[ \binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6 \]

### Step 3: Choose the consonants
Next, we need to choose 3 consonants from the 5 available consonants. The number of ways to choose 3 consonants from 5 is given by:

\[ \text{Number of ways to choose 3 consonants} = \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5!}{3! \cdot 2!} \]

Calculating this gives:

\[ \binom{5}{3} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 4: Calculate the total combinations of vowels and consonants
Now, we multiply the number of ways to choose the vowels and the consonants:

\[ \text{Total combinations} = \binom{4}{2} \times \binom{5}{3} = 6 \times 10 = 60 \]

### Step 5: Arrange the letters
Since we are forming words, we need to arrange the 5 letters (2 vowels + 3 consonants). The number of ways to arrange 5 letters is given by:

\[ \text{Arrangements of 5 letters} = 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

### Step 6: Calculate the total number of words
Finally, the total number of words formed is the product of the total combinations of vowels and consonants and the arrangements of the letters:

\[ \text{Total words} = \text{Total combinations} \times \text{Arrangements} = 60 \times 120 = 7200 \]

### Final Answer
The total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants is **7200**.

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