# In how many ways can the letters of the word ‘PENCIL’

Question:

In how many ways can the letters of the word ‘PENCIL’ be arranged so that N is always next to E?

Solution:

Given: We have 6 letters

To Find: Number of ways to arrange letters $P, E, N, C, I, L$

Condition: $N$ is always next to $E$

Here we need EN together in all arrangements.

So, we will consider EN as a single letter.

Now, we have 5 letters, i.e. $P, C, I, L$ and 'EN'.

5 letters can be arranged in ${ }^{5} P_{5}$ ways

$\Rightarrow{ }^{5} P_{5}$

$\Rightarrow \frac{5 !}{(5-5) !}$

$\Rightarrow \frac{5 !}{0 !}$

$\Rightarrow 120$

In 120 ways we can arrange the letters of the word ‘PENCIL’ so that N is always next to E