Laura and Martin obtain a $20-$year, $$190,000$ conventional mortgage at $10\mathrm{.}5\%$ on a house selling for $$230,000.$ Their monthly mortgage payment, including principal and interest, is $$1896\mathrm{.}20.$

a$)$ Determine the total amount they will pay for their house.

b$)$ How much of the cost will be interest?

c$)$ How much of the first payment on the mortgage is applied to the principal?