Question

If n people including a and b are randomly arranged in a line then what is the probability that the two are next to each other

Answer 1

Answer: $\frac{2}{n}$

n people can be arranged in $n!$ways

But, out of these, a and b are supposed to stand next to each other. So we will consider them as one unit. Therefore, ways in which (n-1) people can stand are $= (n-1)!$

a and b can stand together in two ways: ab or ba

Therefore, the probability that a and b are standing next to each other in a line which has total n people randomly arranged are:

$P=\frac{favorable\hspace{1mm}ways}{Total\hspace{1mm}ways}=\frac{2\times (n-1)!}{n!}=\frac{2 \times (n-1)!}{n\times (n-1)!}=\frac{2}{n}$