Question

The letters of the word PENCIL are arranged in all possible ways. The number of ways in which N always occur next to E is ?

Answer 1

There are $6$ letters in the word $PENCIL$ without any repetition.

Hence, total number of all possible arrangements $\left(N\right)=6!=720.$

Now, take the combination of individual letters $n$ and $e$ as a single letter $n$. So, there are $5$ letters now which can be arranged in $5!=120$ ways.

Again, for each of these arrangements $n$ and $e$ can be arranged in $2!=2$ ways. Hence, total number of arrangements in which $n$ and $e$ will always be next to each other $\left(M\right)=120\ast 2=240.$

Hence, the probability that $n$ and $e$ will always be next to each other$=\frac{M}{N}=\frac{240}{720}=\frac{1}{3}.$

However, if "n is always next to e” means that n will always come after $e$, then the number of cases favourable to this event $=5!=120$ and hence the probablity in this case will be $\frac{120}{720}=\frac{1}{6}.$