To determine the probability that you win the first prize and your mom wins the second prize, let's break down the scenario step by step.
1. Total number of ways to award the prizes:
- There are 10 people in the competition and 10 possible choices for who can win the first prize, 9 choices remaining for who can win the second prize, and 8 choices remaining for who can win the third prize.
- This results in a total of:
[tex]\[
10 \times 9 \times 8 = 720
\][/tex]
ways to award the three prizes.
2. Probability of winning the first prize:
- You are one of the 10 people, so the probability that you win the first prize is:
[tex]\[
\frac{1}{10}
\][/tex]
3. Probability of your mom winning the second prize:
- After you've won the first prize, only 9 people are left for the second prize, one of whom is your mom. So, the probability that your mom wins the second prize, given that you have already won the first prize, is:
[tex]\[
\frac{1}{9}
\][/tex]
4. Combined probability:
- The probabilities are independent events, so the combined probability of both events happening (you winning the first prize and your mom winning the second prize) is the product of the two individual probabilities:
[tex]\[
\frac{1}{10} \times \frac{1}{9} = \frac{1}{90}
\][/tex]
5. Probability in terms of the total ways:
- Given a total of 720 ways to award the prizes, the number of ways in which you win first and your mom wins second is:
[tex]\[
720 \times \frac{1}{90} = 8
\][/tex]
- Therefore, the fraction of the total ways in which you win first and your mom wins second is:
[tex]\[
\frac{8}{720}
\][/tex]
6. Conclusion:
- Thus, the probability that you win first prize and your mom wins second prize can be expressed as:
[tex]\[
\frac{8}{720}
\][/tex]
- Simplifying the fraction does not change its value relevant to the context of this question.
So, the correct answer is:
[tex]\[
\text{C. } \frac{8}{720}
\][/tex]
