Conditional formatting based on rows AND columns

Thanks to both of you!

Using your input I solved it with
OR(COUNTIFS(Person;$A2;C:C;"Full time")>1;COUNTIFS(Person;$A2;C:C;"Part time")>2)
for red formatting.
The rule applies to range $C$2:$E:$100.
Stop if true is checked.
Green formatting is a lot easier with a rule of just "Cell contains" "Full time" or "Part time".

My only remaining problem is that my sheet has become very slow in responding so I am considering a macro, but that should be posted elsewhere, unless anyone here has any tips on speeding it up.
Thanks again,
/Helena