Let's solve the problem step-by-step:
1. Understand the Scenario: There are 8 participants, including you and your mom. There are 3 different prizes to be awarded randomly.
2. Total Number of Ways to Award Prizes: We are given that there are 336 total ways to award the prizes. This means there are 336 different possible combinations for who could win each prize.
3. Specific Event of Interest: We want to find the probability that you win the first prize and your mom wins the second prize.
4. Number of Favorable Outcomes:
- There is exactly 1 specific way in which you can win the first prize and your mom can win the second prize.
- After you win the first prize, there are 7 people left from whom the second-prize winner can be chosen. Once your mom is selected for the second prize, the third prize can be awarded in any manner but it does not affect our current calculation since we are only concerned with the occurrence of you getting the first prize followed by your mom getting the second.
5. Probability Calculation:
- Favorable outcomes: 1 (You win first prize, and your mom wins second)
- Total possible outcomes: 336
Hence, the probability [tex]\( P \)[/tex] is given by:
[tex]\[
P = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{1}{336}
\][/tex]
6. Comparing with the Options:
- A. [tex]$\frac{1}{336}$[/tex]
- B. [tex]$\frac{1}{8}$[/tex]
- C. [tex]$\frac{6}{336}$[/tex]
- D. [tex]$\frac{8}{336}$[/tex]
Based on our calculation, the correct choice is option A: [tex]\( \frac{1}{336} \)[/tex].
Therefore, the probability that you win the first prize and your mom wins the second prize is [tex]\( \boxed{\frac{1}{336}} \)[/tex].
