Let's solve the problem step-by-step.
### Understanding the Problem
There are 9 people, including you and your mom, entering a drawing with 3 different prizes. We are asked to find the probability that you win the first prize and your mom wins the second prize.
### Total Number of Ways to Award the Prizes
We are given that there are 504 ways to award the 3 prizes among the 9 contestants.
### Favorable Outcomes
To find the favorable outcomes where you win the first prize and your mom wins the second prize, we consider:
- The first prize is awarded to you.
- The second prize is awarded to your mom.
There is only one specific way to award the prizes such that you get the first prize and your mom gets the second prize.
### Probability Calculation
The probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
[tex]\[
\text{Probability} = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}
\][/tex]
Given:
- Number of favorable outcomes = 1
- Total number of outcomes = 504
Thus,
[tex]\[
\text{Probability} = \frac{1}{504}
\][/tex]
### Conclusion
The probability that you win the first prize and your mom wins the second prize is:
[tex]\[
\boxed{\frac{1}{504}}
\][/tex]
Therefore, the correct answer is:
[tex]\[
D. \frac{1}{504}
\][/tex]
