jmc

If the co-president positions are unique, there are 15 choices for the first president and 14 choices for the second president. However, since the positions are identical, we must divide by $2$, since $15\cdot 14$ counts each possible pair of co-presidents twice, once for each order in which they are selected. This gives us $\frac{15\times 14}{2}=\boxed{105}$ ways to choose the co-presidents.