To determine the number of ways to assign first, second, and third place among nine students, we need to calculate the number of permutations of 9 students taken 3 at a time.
A permutation considers the order of selection, which is important in this context as the positions (first, second, third) are distinct.
The formula to calculate permutations [tex]\( _nP_r \)[/tex] is given by:
[tex]\[
_nP_r = \frac{n!}{(n-r)!}
\][/tex]
Where:
- [tex]\( n \)[/tex] is the total number of items (in this case, students), which is 9.
- [tex]\( r \)[/tex] is the number of items to choose, which is 3 (for the first, second, and third places).
Applying the values of [tex]\( n \)[/tex] and [tex]\( r \)[/tex]:
[tex]\[
_9P_3 = \frac{9!}{(9-3)!}
\][/tex]
Breaking this down:
[tex]\[
_9P_3 = \frac{9!}{6!}
\][/tex]
Factorials are calculated as the product of all positive integers up to that number. Thus:
[tex]\[
9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1
\][/tex]
[tex]\[
6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1
\][/tex]
For convenience, we simplify the calculation by noticing that [tex]\( 9! \)[/tex] can be written as [tex]\( 9 \times 8 \times 7 \times 6! \)[/tex]. Hence,
[tex]\[
_9P_3 = \frac{9 \times 8 \times 7 \times 6!}{6!} = 9 \times 8 \times 7
\][/tex]
Now, multiplying these together:
[tex]\[
9 \times 8 = 72
\][/tex]
[tex]\[
72 \times 7 = 504
\][/tex]
Thus, the number of ways to assign the first, second, and third place to the nine students is:
[tex]\[
\boxed{504}
\][/tex]
