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

1.	In how many ways can the letters of the word ‘PENCIL’ be arranged so that N is always next to E?
Answer» 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 ⁵P₅ ways ⇒ ⁵P₅ ⇒ \(\frac{5!}{(5-5)!}\) ⇒ \(\frac{5!}{0!}\) ⇒ 120 In 120 ways we can arrange the letters of the word ‘PENCIL’ so that N is always next to E.

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

Answer»

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 ⁵P₅ ways

⇒ ⁵P₅

⇒ \(\frac{5!}{(5-5)!}\)

⇒ \(\frac{5!}{0!}\)

⇒ 120

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