A family buys a house for $$330,000.$ They pay $15\%$ down and the rest is amortized with a $30-$year mortgage at $6\mathrm{.}5\%$ annual interest, compounded monthly.

$($a$)$ What is the nominal amount of the mortgage loan?

$($b$)$ What are the monthly payments?

$($c$)$ What is the total of all the payments over the $30$ years?

(d$)$ What is the total interest paid on the loan over the $30$ years?