To determine the probability of the alarm code beginning with a number greater than 7, let's follow the steps outlined below:
1. Identify Favorable Outcomes:
- Digits greater than 7 are 8 and 9.
- Therefore, there are 2 favorable choices for the first digit.
2. Calculate the Number of Favorable Outcomes:
- After choosing one of the 2 digits (8 or 9) for the first position,
- There are [tex]\(9\)[/tex] remaining digits for the second position.
- For the third position, there are [tex]\(8\)[/tex] remaining digits (excluding the first two chosen digits).
- For the fourth position, there are [tex]\(7\)[/tex] remaining digits.
- Therefore, the number of favorable outcomes is:
[tex]\[ 2 \times 9 \times 8 \times 7 = 2 \times \left(9 P 3\right) \][/tex]
3. Calculate the Total Possible Outcomes:
- There are 10 possible choices for the first digit.
- After choosing the first digit, there are 9 remaining digits for the second position.
- For the third position, there are 8 remaining digits.
- For the fourth position, there are 7 remaining digits.
- Therefore, the number of total possible outcomes is:
[tex]\[ 10 \times 9 \times 8 \times 7 = 10 P 4 \][/tex]
4. Calculate the Probability:
- Probability is the ratio of the number of favorable outcomes to the total number of possible outcomes.
- The probability expression is:
[tex]\[
\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{2 \times 9 P 3}{10 P 4}
\][/tex]
So, the correct expression to determine the probability of the alarm code beginning with a number greater than 7 is:
[tex]\[
\boxed{\frac{\left({ }_2 P_1\right)\left(9 P_3\right)}{10 P_4}}
\][/tex]
