To determine how many ways we can assign first, second, and third place to three students out of nine, we need to calculate the number of permutations. Permutations are used when the order in which we select the students matters, as is the case for assigning different prizes.
Let's break down the calculation step by step:
1. Understand the permutation formula: When selecting [tex]\( r \)[/tex] objects from [tex]\( n \)[/tex] objects, the permutation is given by:
[tex]\[
P(n, r) = \frac{n!}{(n-r)!}
\][/tex]
Here, [tex]\( n \)[/tex] is the total number of students (9), and [tex]\( r \)[/tex] is the number of prizes (3).
2. Substitute the values into the formula:
[tex]\[
P(9, 3) = \frac{9!}{(9-3)!} = \frac{9!}{6!}
\][/tex]
3. Simplify factorials: Recall that [tex]\( n! \)[/tex] (n factorial) is the product of all positive integers up to [tex]\( n \)[/tex].
[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]
We can cancel out the common factorials in the numerator and the denominator:
[tex]\[
\frac{9!}{6!} = \frac{9 \times 8 \times 7 \times 6!}{6!} = 9 \times 8 \times 7
\][/tex]
4. Perform the multiplication:
[tex]\[
9 \times 8 = 72
\][/tex]
[tex]\[
72 \times 7 = 504
\][/tex]
Therefore, there are 504 ways to assign first, second, and third place to three out of nine students.
So, the correct answer is:
[tex]\[
504
\][/tex]
