How many words each of 3 vowels and 2 consonants can be formed from the letters of the word INVOLUTE so that consonants never together?

Answer

To solve the problem of how many words can be formed using 3 vowels and 2 consonants from the letters of the word "INVOLUTE" such that the consonants are never together, we can follow these steps:

### Step 1: Identify the vowels and consonants
In the word "INVOLUTE", we have:
- Vowels: I, O, U, E (Total = 4 vowels)
- Consonants: N, V, L, T (Total = 4 consonants)

### Step 2: Select 3 vowels from the 4 available
We need to choose 3 vowels out of the 4 available vowels. The number of ways to choose 3 vowels from 4 is given by the combination formula:

\[ \text{Number of ways to choose 3 vowels} = \binom{4}{3} = \frac{4!}{3! \cdot 1!} = 4 \]

### Step 3: Select 2 consonants from the 4 available
Next, we need to choose 2 consonants from the 4 available consonants. The number of ways to choose 2 consonants from 4 is given by:

\[ \text{Number of ways to choose 2 consonants} = \binom{4}{2} = \frac{4!}{2! \cdot 2!} = 6 \]

### Step 4: Calculate the total combinations of vowels and consonants
Now, we can find the total combinations of selecting 3 vowels and 2 consonants:

\[ \text{Total combinations} = \text{(Ways to choose vowels)} \times \text{(Ways to choose consonants)} = 4 \times 6 = 24 \]

### Step 5: Arrange the selected letters ensuring consonants are never together
Now, we need to arrange the selected letters (3 vowels and 2 consonants) such that the consonants are never together.

1. First, arrange the 3 vowels. The number of arrangements of 3 vowels is:

\[ 3! = 6 \]

2. Once the vowels are arranged, we can place the consonants. The arrangement of 3 vowels creates 4 possible slots for consonants (before the first vowel, between vowels, and after the last vowel):

- _ V _ V _ V _

3. We need to choose 2 out of these 4 slots to place the consonants. The number of ways to choose 2 slots from 4 is:

\[ \binom{4}{2} = 6 \]

4. The 2 consonants can be arranged in the chosen slots in:

\[ 2! = 2 \]

### Step 6: Calculate the total arrangements
Now, we can calculate the total arrangements:

\[ \text{Total arrangements} = (\text{Arrangements of vowels}) \times (\text{Ways to choose slots}) \times (\text{Arrangements of consonants}) = 6 \times 6 \times 2 = 72 \]

### Final Calculation
Finally, we combine the number of ways to choose the letters and the arrangements:

\[ \text{Total words} = \text{Total combinations} \times \text{Total arrangements} = 24 \times 72 = 1728 \]

### Conclusion
Thus, the total number of words that can be formed with 3 vowels and 2 consonants from the letters of the word "INVOLUTE" such that the consonants are never together is **1728**.