In how many ways, can the letters of the word PENCIL be arranged so that 

1) N is always next to E

2) N and E are always together.

Hi!
Here is the answer to your question.
 
There are 6 letters in the word ‘PENCIL’.
(1)
Consider EN as one letter. Now 5 letters (P, C, I, L, EN) can be arranged in 5P5 = 5! = 120 ways.
Hence, total number of ways in which N is always next to E is 120.
 
(2)
Consider EN as one letter. Now, 5 letters can be arranged in 5P5 = 5! = 120 ways
E and N can arrange among themselves in 2! = 2 ways.
Hence, the total number of ways in which N and E are always together = 120 × 2 = 240.
 
Cheers!

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