The number of five-letter words containing 3 vowels and 2 consonants that can be formed using the letters of the word 'EQUATION' so that the two consonants occur together, is

Options

A. 720
B. 1440
C. 240
D. 480

Explanation

To solve the problem of finding the number of five-letter words containing 3 vowels and 2 consonants from the letters of the word "EQUATION," where the two consonants must occur together, we can follow these steps:

### Step 1: Identify the Vowels and Consonants
The letters in the word "EQUATION" consist of:
- **Vowels**: E, U, A, I, O (5 vowels)
- **Consonants**: Q, T, N (3 consonants)

### Step 2: Choose the Vowels and Consonants
We need to select:
- 3 vowels from the 5 available vowels
- 2 consonants from the 3 available consonants

The number of ways to choose the vowels is given by the combination formula \( nCr \):
- Choosing 3 vowels from 5: \( \binom{5}{3} \)
- Choosing 2 consonants from 3: \( \binom{3}{2} \)

Calculating these:
\[ \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]
\[ \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3}{1} = 3 \]

### Step 3: Treat Consonants as a Single Unit
Since the two consonants must occur together, we can treat them as a single unit or block. This gives us:
- 3 vowels (V1, V2, V3)
- 1 block of consonants (C1C2)

This means we now have 4 units to arrange: V1, V2, V3, and (C1C2).

### Step 4: Arrange the Units
The number of ways to arrange these 4 units is given by \( 4! \):
\[ 4! = 24 \]

### Step 5: Arrange the Consonants Within Their Block
The two consonants can be arranged among themselves in \( 2! \) ways:
\[ 2! = 2 \]

### Step 6: Calculate the Total Number of Arrangements
Now, we can calculate the total number of arrangements by multiplying the number of ways to choose the vowels, the number of ways to choose the consonants, the arrangements of the units, and the arrangements of the consonants within their block:
\[ \text{Total arrangements} = \binom{5}{3} \times \binom{3}{2} \times 4! \times 2! \]
Substituting the values we calculated:
\[ \text{Total arrangements} = 10 \times 3 \times 24 \times 2 \]

Calculating this:
\[ 10 \times 3 = 30 \]
\[ 30 \times 24 = 720 \]
\[ 720 \times 2 = 1440 \]

### Final Answer
The total number of five-letter words containing 3 vowels and 2 consonants, where the consonants occur together, is **1440**.

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