Question: 2015 Winter Final - 15
Let $S$ be a set of 100 integers; 30 of these are positive and the other 70 integers in $S$ are
negative. We choose, uniformly at random, a 20-element subset of $S$. Let $x$ be the product of the
integers in the chosen subset. What is the probability that $x > 0$?
(a)
$\frac{1}{{100 \choose 20}} \sum_{k=0}^{20} \left({70 \choose k} + {30 \choose 20 - k}\right)$
(b)
$\frac{1}{{100 \choose 20}} \sum_{k=0}^{10} \left({70 \choose 2k} + {30 \choose 20 - 2k}\right)$
(c)
$\frac{1}{{100 \choose 20}} \sum_{k=0}^{10} {70 \choose 2k}{30 \choose 20 - 2k}$
(d)
$\frac{1}{{100 \choose 20}} \sum_{k=0}^{20} {70 \choose k}{30 \choose 20 - k}$