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

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 in the word 'EQUATION'.
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 3 vowels from the 5 available vowels.
We need to calculate the number of ways to choose 3 vowels from the 5 vowels. This can be done using the combination formula:
\[ \text{Number of ways to choose 3 vowels} = \binom{5}{3} \]
Calculating this:
\[ \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5!}{3! \cdot 2!} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 3: Choose 2 consonants from the 3 available consonants.
Next, we need to calculate the number of ways to choose 2 consonants from the 3 consonants:
\[ \text{Number of ways to choose 2 consonants} = \binom{3}{2} \]
Calculating this:
\[ \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3!}{2! \cdot 1!} = \frac{3}{1} = 3 \]

### Step 4: Calculate the total combinations of chosen vowels and consonants.
Now, we multiply the number of ways to choose the vowels and consonants:
\[ \text{Total combinations} = \binom{5}{3} \times \binom{3}{2} = 10 \times 3 = 30 \]

### Step 5: Arrange the selected vowels and consonants.
We have selected a total of 5 letters (3 vowels and 2 consonants). The number of ways to arrange these 5 letters is given by:
\[ \text{Arrangements} = 5! = 120 \]

### Step 6: Calculate the total number of words formed.
Finally, the total number of distinct words that can be formed is:
\[ \text{Total words} = \text{Total combinations} \times \text{Arrangements} = 30 \times 120 = 3600 \]

### Final Answer:
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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