The young man and the young woman paid ------ attention to t

not any= no没错,但是这个not 是指 将动词变成否定,即：
The young man and the young woman didn't pay any attention.
=paid no attention.
其他例子还有：
I have no pencils= I don't have any pencils.
通过以上的分析楼主应该可以看出来,由于题目中前面的句子已经给定了,所以Not any 不能选.