You are considering a mortgage for $30$ years with initial value $$280,000,$ and annual interest rate $4\mathrm{.}75\%.$ You can reduce your interest rate by buying points. The bank will sell you a point for $1\%$ of the initial value of the loan. Each point you buy reduces your interest rate by $0\mathrm{.}45\%.$ You are considering buying $5$ points. If you buy these points, how long until your lower monthly payments pay back the cost for these points?

a$)21$ months.

b$)24$ months.

c$)28$ months.

d$)32$ months.

e$)37$ months.

f$)40$ months.