A trader creates a strangle by buying a $1-$month call with a strike of $$30$ for $$3$ and buying a $1-$month put with a strike of $$25$ for $$2.$

a$.$ What is the initial investment of the strategy?

b$.$ What is the total payoff $($excluding the initial investment$)$ when the stock price in one month is $($i$)$ $$20,($ii$)$ $$27\mathrm{.}5,$ and $($iii$)$ $$35.$