To solve the problem of finding the number of nine-lettered words that can be formed using all the letters of the word "MEENANSHU" such that alike letters are never adjacent, we can follow these steps:
### Step 1: Identify the Letters and Their Frequencies
The word "MEENANSHU" consists of the following letters:
- M: 1
- E: 2
- N: 2
- A: 1
- S: 1
- H: 1
- U: 1
### Step 2: Calculate the Total Arrangements Without Restrictions
The total number of arrangements of the letters in "MEENANSHU" can be calculated using the formula for permutations of multiset:
\[
\text{Total arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!}
\]
where \( n \) is the total number of letters and \( p_i \) are the frequencies of the repeated letters.
Here, \( n = 9 \) (total letters), \( p_1 = 2 \) (for E), and \( p_2 = 2 \) (for N):
\[
\text{Total arrangements} = \frac{9!}{2! \times 2!} = \frac{362880}{4} = 90720
\]
### Step 3: Calculate Arrangements Where Alike Letters Are Together
We will use the principle of complementary counting. We first calculate the arrangements where the alike letters (E's and N's) are together.
#### Case 1: E's Together
If we treat the two E's as a single unit, we have the following letters to arrange: (EE), M, N, N, A, S, H, U. This gives us 8 units to arrange, where N is repeated twice:
\[
\text{Arrangements with E's together} = \frac{8!}{2!} = \frac{40320}{2} = 20160
\]
#### Case 2: N's Together
If we treat the two N's as a single unit, we have the following letters to arrange: M, E, E, (NN), A, S, H, U. This also gives us 8 units to arrange, where E is repeated twice:
\[
\text{Arrangements with N's together} = \frac{8!}{2!} = 20160
\]
#### Case 3: Both E's and N's Together
If we treat both E's and N's as single units, we have the following letters to arrange: (EE), (NN), M, A, S, H, U. This gives us 7 distinct units:
\[
\text{Arrangements with both E's and N's together} = 7! = 5040
\]
### Step 4: Apply Inclusion-Exclusion Principle
Now, we apply the inclusion-exclusion principle to find the total arrangements where either E's or N's are together:
\[
\text{Total with E's or N's together} = (\text{E's together}) + (\text{N's together}) - (\text{Both together})
\]
\[
= 20160 + 20160 - 5040 = 35280
\]
### Step 5: Calculate Arrangements Where Alike Letters Are Not Together
Finally, we subtract the arrangements where the alike letters are together from the total arrangements:
\[
\text{Arrangements where no alike letters are together} = \text{Total arrangements} - \text{Total with E's or N's together}
\]
\[
= 90720 - 35280 = 55440
\]
### Final Answer
Thus, the number of nine-lettered words that can be formed using all the letters of "MEENANSHU" such that alike letters are never adjacent is **55440**.
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