In how many ways can the letters of the word 'PARALLEL' be arranged so that all L' s do not come together ?

Explanation

To solve the problem of arranging the letters of the word "PARALLEL" such that all L's do not come together, we can follow these steps:

### Step 1: Count the total arrangements of the letters in "PARALLEL".
The word "PARALLEL" consists of 8 letters: P, A, R, A, L, L, E, L.

- The letters consist of:
- P: 1
- A: 2
- R: 1
- L: 3
- E: 1

The formula for the total arrangements of letters when some letters are repeated is given by:

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

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

In this case:

\[ n = 8, \quad p_A = 2, \quad p_L = 3 \]

Thus, the total arrangements can be calculated as:

\[ \text{Total arrangements} = \frac{8!}{2! \times 3!} \]

Calculating this gives:

\[ 8! = 40320, \quad 2! = 2, \quad 3! = 6 \]

So,

\[ \text{Total arrangements} = \frac{40320}{2 \times 6} = \frac{40320}{12} = 3360 \]

### Step 2: Count the arrangements where all L's are together.
To find the arrangements where all L's are together, we can treat the three L's as a single entity or block.

Now, we have the following letters to arrange: P, A, R, A, (LLL). This gives us 6 entities to arrange: P, A, R, A, and the block LLL.

The new arrangement consists of:

- P: 1
- A: 2
- R: 1
- LLL: 1 (treated as one letter)

Using the same formula:

\[ \text{Arrangements with L's together} = \frac{6!}{2!} \]

Calculating this gives:

\[ 6! = 720, \quad 2! = 2 \]

So,

\[ \text{Arrangements with L's together} = \frac{720}{2} = 360 \]

### Step 3: Calculate the arrangements where L's do not come together.
To find the arrangements where the L's do not come together, we subtract the arrangements where the L's are together from the total arrangements:

\[ \text{Arrangements where L's do not come together} = \text{Total arrangements} - \text{Arrangements with L's together} \]

Substituting the values we calculated:

\[ \text{Arrangements where L's do not come together} = 3360 - 360 = 3000 \]

### Final Answer
Thus, the number of ways the letters of the word "PARALLEL" can be arranged such that all L's do not come together is **3000**.

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