1. **Stating the problem:** A school has a total of 78 possible pairs of students. We need to find how many students are participating. 2. **Formula used:** The number of ways to form pairs (2 students) from $n$ students is given by the combination formula: $$\binom{n}{2} = \frac{n(n-1)}{2}$$ 3. **Set up the equation:** Since the total pairs are 78, $$\frac{n(n-1)}{2} = 78$$ 4. **Solve for $n$:** Multiply both sides by 2: $$n(n-1) = 156$$ 5. Expand: $$n^2 - n = 156$$ 6. Rearrange to standard quadratic form: $$n^2 - n - 156 = 0$$ 7. Use the quadratic formula: $$n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-1$, $c=-156$. 8. Calculate the discriminant: $$\sqrt{(-1)^2 - 4 \times 1 \times (-156)} = \sqrt{1 + 624} = \sqrt{625} = 25$$ 9. Find the roots: $$n = \frac{1 \pm 25}{2}$$ 10. Possible values: $$n = \frac{1 + 25}{2} = 13$$ $$n = \frac{1 - 25}{2} = -12$$ (discard negative) 11. **Answer:** There are $\boxed{13}$ students participating in the activity.