Convert 200 mg to g.
$"1000 mg=1 g"$
$200cancel"mg"xx(1"g")/(1000cancel"mg")="0.2 g"$
Convert carats to grams to moles.
$"1.00 ct=0.2 g"$
$1"mol C=12.0107 g C"$
$1.00cancel"ct C"xx(0.2cancel"g C")/(1cancel"ct C")xx(1"mol C")/(12.0107cancel"g C")="0.02 mol C"$
Convert moles C to atoms C.
$1 "mol C"=6.022xx10^23 "atoms C"$
$0.02cancel"mol C"xx(6.022xx10^23"atoms C")/(1cancel"mol C")=1xx10^22" atoms C"$ (rounded to one significant figure)