$\frac25\times\frac34+\frac35\times\frac24=\frac{3}{5}$ 
Two different numbers are randomly selected from the set $-5, -3, -1, 3, 5$~and multiplied together. What is the probability that the product is a negative number? (adapted from $2016$ AMC $8$ Problem, Question \#$13$) 
Options:
1. $\frac{5}{6}$ 
2. $\frac{2}{5}$ 
3. $\frac{3}{4}$ 
4. $\frac{1}{2}$ 
5. $\frac{3}{5}$ 