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To bomb a district to smithereens means that the government is punishing the people of the district for whatever "sins" they have committed against the government. Katniss has come from District 13 originally and now lives in her beautiful house provided because she won the Hunger Games.
District 13 staged an ill-fated rebellion against the Capitol, the end result being that District 13 was wiped out. The District however is said to still be alive.
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