$7.$ Consider a $16*16$ Banyan network. Two cells enter the switch and are to be switched during the same switching cycle.

$($a$)[5$ Marks$]$ Schedule the $16*16$ Banyan Network.

$($b$)[6$ Marks$]$ Determine the probability that there will be blocking if the output destinations are randomly selected from the $16$ outputs with each output being equally likely, assuming that

$($i$)$ the two cells arrive at inputs $4$ and $5,$

$($ii$)$ the two cells arrive at inputs $0$ and $11,$ and

$($iii$)$ the two cells arrive at any two inputs. $($c$)[4$ Marks$]$ Assume now that the two cells come into the switch destined for outputs $4(100)$ and $5(101).$ Determine the probability, across all possible input combinations for the $2$ cells, that there will be blocking.