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: Count the total permutations of the letters in "ENSHRINE".

The word "ENSHRINE" consists of 9 letters where the letters 'E' and 'N' are repeated twice. The formula for calculating the total permutations of a word with repeated letters is given by:

\[ \text{Total permutations} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!} \]

where \( n \) is the total number of letters, and \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeated letters.

For "ENSHRINE":
- Total letters, \( n = 9 \)
- Frequency of 'E' = 2
- Frequency of 'N' = 2

Thus, the total permutations are:

\[ \text{Total permutations} = \frac{9!}{2! \times 2!} = \frac{362880}{4} = 90720 \]

### Step 2: Calculate the permutations where the two 'E's are together.

If we consider the two 'E's as a single unit (block), we effectively reduce the number of letters to arrange. The letters to arrange are now: (EE), N, S, H, R, I, N, which gives us 8 units in total.

The frequency of 'N' is still 2. Thus, the permutations where the two 'E's are together is:

\[ \text{Permutations with E together} = \frac{8!}{2!} = \frac{40320}{2} = 20160 \]

### Step 3: Calculate the permutations where the two 'N's are together.

Similarly, if we consider the two 'N's as a single unit (block), we have the letters: E, E, (NN), S, H, R, I, which gives us 8 units in total.

The frequency of 'E' is still 2. Thus, the permutations where the two 'N's are together is:

\[ \text{Permutations with N together} = \frac{8!}{2!} = \frac{40320}{2} = 20160 \]

### Step 4: Calculate the permutations where both 'E's and 'N's are together.

Now, if we consider both pairs (EE and NN) as single units, we have the letters: (EE), (NN), S, H, R, I, which gives us 7 units in total.

Thus, the permutations where both 'E's and 'N's are together is:

\[ \text{Permutations with both E and N together} = 7! = 5040 \]

### Step 5: Apply the principle of inclusion-exclusion.

To find the number of arrangements where no two alike letters are together, we use the principle of inclusion-exclusion:

\[ \text{Valid arrangements} = \text{Total arrangements} - \text{(E together)} - \text{(N together)} + \text{(Both E and N together)} \]

Substituting the values we calculated:

\[ \text{Valid arrangements} = 90720 - 20160 - 20160 + 5040 \]

Calculating this gives:

\[ \text{Valid arrangements} = 90720 - 40320 + 5040 = 55740 \]

### Final Answer:
The number of permutations of the letters of the word "ENSHRINE" in which no two alike letters are together is **55740**.