The smallest product one could obtain by multiplying two numbers in the set $\{ -7,-5,-1,1,3 \}$ is
$\text{(A)}\ -35 \qquad \text{(B)}\ -21 \qquad \text{(C)}\ -15 \qquad \text{(D)}\ -1 \qquad \text{(E)}\ 3$
To get the smallest possible product, we want to multiply the smallest negative number by the largest positive number.  These are $-7$ and $3$, respectively, and their product is $-21$, which is $\boxed{\text{B}}$