# 2017-22 Debugging

Let $$p$$, $$q$$, $$r$$ be positive integers such that $$p,q\ge r$$. Ada and Betty independently read all source codes of their programming project. Ada found $$p$$ bugs and Betty found $$q$$ bugs, including $$r$$ bugs that Ada found. What is the expected number of remaining bugs that neither Ada nor Betty found?

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Let $$n$$ be a fixed positive integer and let $$p\in (0,1)$$. Let $$D_n$$ be the determinant of a random $$n\times n$$ 0-1 matrix whose entries are independent identical random variables, each of which is 1 with the probability $$p$$ and 0 with the probability $$1-p$$.  Find the expected value and variance of $$D_n$$.