To determine the number of ways to choose the first and second students in the spelling bee from a group of 10 students, we treat this as a permutation problem because the order in which the students are chosen matters (the student who goes first is different from the one who goes second).
We can calculate the permutations using the formula for permutations, given as:
\[ P(n, r) = \frac{n!}{(n - r)!} \]
Here:
- \(n\) is the total number of students (10),
- \(r\) is the number of positions to fill (2).
Plugging in the values, we get:
\[ P(10, 2) = \frac{10!}{(10 - 2)!} = \frac{10!}{8!} \]
Now, simplifying this, we can cancel \(8!\):
\[ \frac{10 \times 9 \times 8!}{8!} = 10 \times 9 = 90 \]
Therefore, the number of ways the students can be chosen for the first and second spots in the spelling bee is \( \boxed{90} \).