How many words can be formed with 3 vowels and 2 consonants taken from the word 'EQUATION'?

Options

A. `30`
B. `120`
C. `1200`
D. `3600`

Answer

To solve the problem of how many words can be formed with 3 vowels and 2 consonants taken from the word "EQUATION", we can follow these steps:

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

### Step 2: Choose the Vowels
We need to choose 3 vowels from the 5 available vowels. The number of ways to choose 3 vowels from 5 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

\[ \text{Number of ways to choose 3 vowels} = \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 3: Choose the Consonants
Next, we need to choose 2 consonants from the 3 available consonants.

\[ \text{Number of ways to choose 2 consonants} = \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3}{1} = 3 \]

### Step 4: Calculate the Total Combinations
Now, we multiply the number of ways to choose the vowels by the number of ways to choose the consonants.

\[ \text{Total combinations of choosing vowels and consonants} = \binom{5}{3} \times \binom{3}{2} = 10 \times 3 = 30 \]

### Step 5: Arrange the Chosen Letters
After selecting 3 vowels and 2 consonants, we have a total of 5 letters to arrange. The number of ways to arrange 5 letters is given by \( 5! \).

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

### Step 6: Calculate the Total Words Formed
Finally, we multiply the total combinations of choosing the letters by the number of arrangements.

\[ \text{Total number of words} = \text{Total combinations} \times \text{Arrangements} = 30 \times 120 = 3600 \]

### Final Answer
Thus, the total number of words that can be formed with 3 vowels and 2 consonants taken from the word "EQUATION" is **3600**.

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