Consider three different processors: P$1,$ P$2,$ and P$3$ executing the same instruction set. P$1$ has a $3\mathrm{.}2$ GHz clock rate, a CPI $($cycles per instruction$)$ of $1\mathrm{.}5,$ P$2$ has a $4\mathrm{.}3$ GHz clock rate and a CPI of $2,$ and P$3$ has a $5\mathrm{.}1$ GHz clock rate and a CPI of $1\mathrm{.}4.$

a$.$ Which processor has the highest performance measured in instructions per second? Show your calculations.

b$.$ If the processors each execute a program in $20$ seconds, then find the number of cycles and the number of instructions. Show your calculations.