A product is made of an assembly of $10$ parts, each produced at a different workcell. The$$average defect rate of the parts before inspection in each cell is $5\%.$ Before leaving the cell,$$every part passes through a two$-$stage inspection process, each stage of which identifies$95\%$ of the remaining defects. No identified defects are allowed to proceed without being$$corrected$.$a$.$ What is the defect rate of parts following the inspections? What is this rate expressed as ppm$?$b$.$ Suppose a defect in any of the $10$ parts results in a defective assembled product. What is$$the defect rate of assembled products in ppm$?$c$.$ If a mistake occurs while assembling $1$ in every $1000$ products $($assuming the assembly$$uses the $10$ parts described earlier$),$ what then is the defect rate in ppm$?$d$.$ Assume the maximum acceptable defect rate is $250$ ppm$.$ Assuming no change in the$$quality of the $10$ parts used in assembly, what must the mistake rate during assembly be$$reduced to$?$