$($i$)$ In how many ways can we arrange three A$\text{'}$s$,$ two B$\text{'}$s and five C$\text{'}$s in

series;

$($ii$)$ In how many ways can we arrange two A$,$ four B and three C in

order so that the two A$\text{'}$s are consecutive?

$($iii$)$ In how many ways can we arrange five A$\text{'}$s$,$ two B$\text{'}$s and six C$\text{'}$s in a row

so that the B$\text{'}$s have exactly $3$ other letters between them?

$($iv$)$ In how many ways can we arrange three A$\text{'}$s$,$ four B$\text{'}$s and five C$\text{'}$s in

order so that the C$\text{'}$s are not all consecutive?

$($v$)$ In how many ways can we arrange ten A$\text{'}$s$,$ six B$\text{'}$s and five C$\text{'}$s in a row so that there are no two B$\text{'}$s in

consecutive positions?