The Allies avoid a costly invasion of the Japanese home islands by: A. Signing a peace treaty with Japan.
B. Crushing the Japanese navy.
C. Dropping the atomic bomb.
D. Killing the Emperor.

C. Dropping the atomic bomb

Explanation:

The allies had already fought the hard island hopping campaign, and they knew an invasion of Japan would be very hard. They also saw that the Soviet Union rather than liberating eastern Europe it was occupying and establish communist regimes. This caused the US and UK to come to an agreement and drop 2 bombs on Japan, on August 6th, Hiroshima, on the 9th, Nagasaki. WW2 ended immediately as the Japanese emperor surrendered.

fairly positive it is 4