You have left off the quotation marks, so that contextIntegerFunctions.pl becomes the two string constants "contextIntergetFunction" and "pl" combined by the dot operation, which is string concatenation, and so you get "contextIntegerFunctionspl", with no dot. Notice that that is the filename that appears in the error message.
PS, if the penalty in a Panic::Button is 0, you don't need to specify it as the default is 0.
PPS, note that contextIntegerFunctions.pl is needed only if you want the STUDENT to be able to enter C(n,m) and P(n,m) in their answers. If you only need them for the author, they are already part of PGauxiliaryFunctions.pl, which is loaded automatically by PGstandard.pl. If you want the students to compute the numeric values of these functions, don't use contextIntegerFunctions.pl.