We do not allow ________ in the lecture hall,so you are not

这道题选A
前一个空是因为allow doing sth,而后一个空并不是加什么doing或to do,那只不过是一般现在时的被动语态 be allowed 偷 do sth
至于你所说的allow doing与to do如何区分 也送给你吧
allow既可作及物动词又可作不及物动词(多用于及物动词),
1）allow sb.to do sb.允许某人去做某事,
The doctor allowed him to smoke.医生允许他吸烟.
2）allow sb doing sth.允许某人做某事,可以表示当前可允许做的事或习惯性动作.根据上下文有时sb可以省略.如：
Is not allow (anyone)smoking in this meeting room.这间会议室里是不允许吸烟的.
类似的词还有:advise,forbid,permit,encourage等,如:
3) We advised him to have a good rest.我们劝他好好休息.
4) He advised going to Beijing for the holiday.他建议到北京去度假.