$2.$Let n be a positive integer. Consider the random experiment in which weselect a uniformly random permutation $\backslash$pi of $\{1,...,$n$\}$ and then successively insert keys $\backslash$pi $(1),\backslash$pi $(2)...,\backslash$pi $($n$)$ into an $($initially empty$)$ ordinary binary search tree $($BST$).$ Let T denote the resultingBST. In the following parts, for any i in $\{1,...,$ nj$\},$ we write $$node i$$ to refer to the node with key i in T$.$

$($a$)$ Let i belong to $\{1,...,$n$-1\}.$ Explain why the probability that node i is adescendant of node i$+1$ is exactly $1/2.$

$($b$)$Let i and j be distinct elements of $\{1...,$n$\}.$ Give an exact closed$-$form expression for the probability that node i is a descendant of node j$.$ Briefly justify your answer.