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2007 AMC 12A Problems/Problem 12

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Problem

Integers a, b, c, and d, not necessarily distinct, are chosen independantly and at random from 0 to 2007, inclusive. What is the probability that ad-bc is even?

Solution

The only times when ad-bc is even is when ad and bc are of the same parity. The chance of ad being odd (1/2)*(1/2)=1/4, and has a 3/4 probability that it will be even. Therefore, the probability that ad-bc will be even is (1/4)^2+(3/4)^2=5/8

See also

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