A sample of poly$($styrene$)($PS$)$ was found to have three different polymer chain

lengths in the following proportions: $32\%7$ x $105$g mol$-1,20\%6$ x $105$g mol$-1,$ and $10\%4$ x $105$g mol$-1.$ Calculate the following:

$($i$)$ Number average molecular weight.

$($ii$)$ Weight average molecular weight.

$($iii$)$ Polydispersity Index $($PDI$).$

Is the polymer material biological or synthetic in origin? Explain.