In how many ways can the letters of the word “PENCIL” be arranged so that I is always next to L.

Answer

To find the number of ways to arrange the letters of the word "PENCIL" such that the letters 'I' and 'L' are always next to each other, we can follow these steps:

### Step 1: Treat 'I' and 'L' as a single unit
Since 'I' must always be next to 'L', we can consider them as a single unit or block. We can denote this block as 'IL'.

### Step 2: Count the total units
Now, instead of the 6 individual letters (P, E, N, C, I, L), we have the following units:
- The block 'IL'
- The letters P, E, N, and C

This gives us a total of 5 units to arrange: {IL, P, E, N, C}.

### Step 3: Calculate the arrangements of the units
The number of ways to arrange these 5 units is given by the factorial of the number of units:
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

### Step 4: Consider the arrangement of 'I' and 'L' within their block
Since 'I' and 'L' can be arranged in 2 different ways (IL or LI), we need to multiply the number of arrangements of the units by the arrangements of 'I' and 'L' within their block:
\[ 2! = 2 \times 1 = 2 \]

### Step 5: Calculate the total arrangements
Now, we multiply the arrangements of the units by the arrangements of 'I' and 'L':
\[ 120 \times 2 = 240 \]

### Final Answer
Thus, the total number of ways to arrange the letters of the word "PENCIL" such that 'I' is always next to 'L' is **240**.

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