Why did they let the Holocaust happen?

The 'world' didn't care.

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A lot of the world didn't know the situation was as bad as it was in Europe during that time.

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Actually, the Holocaust was regularly reported in the media in Allied and neutral countries from November 1942 on, but seems not to have registered widely. Part of the trouble was that in practical terms there was very little that the Allies could do about it - and of course many people just didn't care.