The number of permutations of alphabets of the word ''ENSHRINE'' in which no two alike alphabets are are together is equal to

Answer

To find the number of permutations of the letters in the word "ENSHRINE" such that no two alike letters are together, we can follow these steps:

### Step 1: Identify the letters and their frequencies
The word "ENSHRINE" consists of the following letters:
- E: 2
- N: 2
- S: 1
- H: 1
- R: 1
- I: 1

### Step 2: Calculate the total permutations of the letters
The total number of letters in "ENSHRINE" is 8. The formula for permutations of a multiset is given by:

\[ \text{Total permutations} = \frac{n!}{n_1! \times n_2! \times \ldots \times n_k!} \]

Where:
- \( n \) is the total number of letters,
- \( n_1, n_2, \ldots, n_k \) are the frequencies of the distinct letters.

For "ENSHRINE", we have:

\[ \text{Total permutations} = \frac{8!}{2! \times 2!} \]

Calculating this:

\[ 8! = 40320 \]
\[ 2! = 2 \]
Thus,

\[ \text{Total permutations} = \frac{40320}{2 \times 2} = \frac{40320}{4} = 10080 \]

### Step 3: Calculate permutations where alike letters are together
To find the arrangements where the two E's and two N's are together, we can treat the pairs (EE) and (NN) as single units. This gives us the following units to arrange:
- (EE), (NN), S, H, R, I

This results in 6 units to arrange. The number of arrangements of these 6 units is:

\[ 6! = 720 \]

### Step 4: Calculate permutations where alike letters are NOT together
To find the arrangements where no two alike letters are together, we subtract the arrangements where alike letters are together from the total arrangements:

\[ \text{Permutations where alike letters are NOT together} = \text{Total permutations} - \text{Permutations where alike letters are together} \]

Substituting the values we calculated:

\[ \text{Permutations where alike letters are NOT together} = 10080 - 720 = 9360 \]

### Final Answer
The number of permutations of the letters in the word "ENSHRINE" such that no two alike letters are together is **9360**.

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