To solve this problem, we need to calculate how many words can be formed using 3 consonants and 2 vowels, with the condition that no two consonants are adjacent.

Steps:

1. Choosing consonants and vowels:

   • Let the total number of consonants be $C$, and the total number of vowels be $V$.
   • First, we select 3 consonants from $C$ available consonants, and 2 vowels from $V$ available vowels.
   • The number of ways to select 3 consonants from $C$ is given by $\binom{C}{3}$, and the number of ways to select 2 vowels from $V$ is given by $\binom{V}{2}$.

2. Arranging the vowels and consonants:

   • There are 2 vowels, and we need to arrange them in such a way that no two consonants are adjacent. To do this, we first arrange the vowels.
   • The number of arrangements for 2 vowels is $2!$.

3. Placing the consonants:

   • Once the vowels are arranged, there are 3 positions available (gaps between the vowels and before/after them) where we can place the 3 consonants.
   • The number of ways to place 3 consonants in these gaps is $3!$.

Thus, the total number of words that can be formed is the product of:

• The number of ways to choose 3 consonants and 2 vowels.
• The number of ways to arrange the vowels.
• The number of ways to arrange the consonants.

Formula:

$\text{Total number of words} = \binom{C}{3} \times \binom{V}{2} \times 2! \times 3!$

Please provide the total number of consonants ($C$) and vowels ($V$), and I can help you compute the result!

Would you like a detailed breakdown of any part, or have any specific questions?

Related Questions:

1. How would the answer change if we allowed consonants to be adjacent?
2. What happens if the number of consonants and vowels were equal?
3. How many words can be formed with 4 consonants and 3 vowels?
4. How does the solution change if we can use each consonant or vowel more than once?
5. Can you derive a general formula for any number of consonants and vowels?

Tip:

When solving problems involving arrangement, always consider how constraints (like "no two items adjacent") affect the placement choices.