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Sentence goes like this:
The man goes through airport screening and manages to persuade security that his biological samples--one for each of the many cities on his itinerary--are harmless
What does mean by one for each of the many cities on his itinerary--are harmless?
The double hyphen -- represents a dash [1.][2.]. Words between dashes give information about the sentence in which it is embedded. For the moment, let us ignore that information and focus on the rest:
The man goes through airport screening and manages to persuade security that his biological samples are harmless.
It should be clear now that the word harmless refers to the biological samples. Dangerous biological materials are usually referred to as biological hazards because they are "biological substances that pose a threat to the health of living organisms, primarily that of humans". So, the security at the airport believed that his biological samples were biohazards and are not allowed on airplanes. However, he was able to convince them that the materials were safe.
Now we return to the dashed part:
one for each of the many cities on his itinerary
Definition:
itinerary
the places you go to or plan to go to on a journey
So, the man took a sample at each city that he visited, every city on his list (itinerary). So all together, he took one sample from each city on his list, and there were many cities on that list.
It means the man was on trip visiting many different cities, and at each city, he took a sample of something biological, which means it's alive and he had a lot of them.
People are not allowed to carry toxic or infectious biological substances on a commercial flight
so the man needed to convince TSA that what he was carrying was safe.
An itinerary is a list of places to visit. So the man was going from city to city with his biological substances. And in each city he convinced airport security that the biological substances he had with him were safe. Although they might not be.
