$($b$)$ A binary number consists only of the digits $0$ and $1.$

$($i$)$ How many different nine$-$digit binary numbers are there?

$($ii$)$ How many different nine$-$digit binary numbers are there which contain exactly four $1s?$

$(2)$

$(3)$

$($iii$)$ How many different nine$-$digit binary numbers are there which contain exactly four $15$ which are all adjacent $($next to each other$)?$