To solve the problem of finding the number of different words that can be formed using all the letters of the word 'SHASHANK' such that the vowels are separated by at least two consonants, we can follow these steps:
### Step 1: Identify the letters and their counts
The word 'SHASHANK' consists of 8 letters:
- Vowels: A, A (2 A's)
- Consonants: S, H, S, H, N, K (6 consonants: S appears twice, H appears twice)
### Step 2: Arrange the consonants
First, we need to arrange the consonants. The consonants are S, H, S, H, N, K. The total arrangements of these consonants can be calculated using the formula for permutations of multiset:
\[
\text{Total arrangements of consonants} = \frac{n!}{p_1! \times p_2!}
\]
Where:
- \( n \) is the total number of consonants,
- \( p_1, p_2, \ldots \) are the frequencies of the repeating consonants.
Here, \( n = 6 \) (total consonants), \( p_1 = 2 \) (for S), \( p_2 = 2 \) (for H).
\[
\text{Total arrangements of consonants} = \frac{6!}{2! \times 2!} = \frac{720}{2 \times 2} = \frac{720}{4} = 180
\]
### Step 3: Position the vowels
Next, we need to position the vowels (A, A) such that they are separated by at least two consonants.
When we arrange the 6 consonants, they create 7 possible slots for placing the vowels (before the first consonant, between consonants, and after the last consonant):
- _ C _ C _ C _ C _ C _ C _
Where C represents a consonant.
### Step 4: Choose slots for the vowels
We need to choose 2 slots from these 7 available slots such that there are at least 2 consonants between the vowels.
To ensure that the vowels are separated by at least two consonants, we can visualize it as follows:
- If we place the first vowel in one of the slots, the next vowel can only be placed in the slots that are at least two positions away.
For example, if we place the first A in slot 1, the second A can be placed in slots 4, 5, 6, or 7.
Calculating the valid combinations:
1. If the first A is in slot 1, valid positions for the second A are 4, 5, 6, 7 (4 choices).
2. If the first A is in slot 2, valid positions for the second A are 5, 6, 7 (3 choices).
3. If the first A is in slot 3, valid positions for the second A are 6, 7 (2 choices).
4. If the first A is in slot 4, valid position for the second A is 7 (1 choice).
Adding these choices gives us:
\[
4 + 3 + 2 + 1 = 10
\]
### Step 5: Calculate total arrangements
Now, we can calculate the total number of arrangements by multiplying the arrangements of consonants by the arrangements of vowels:
\[
\text{Total arrangements} = \text{(Ways to arrange consonants)} \times \text{(Ways to place vowels)}
\]
\[
\text{Total arrangements} = 180 \times 10 = 1800
\]
### Final Answer
Thus, the total number of different words that can be formed using all the letters of the word 'SHASHANK' such that the vowels are separated by at least two consonants is **1800**.
