Let's break down the problem step-by-step to find the probability of a four-digit alarm code beginning with the digit 7.
### Step-by-Step Solution
1. Understanding the total number of four-digit codes:
- Each digit in a four-digit code can be any number from 0 to 9.
- Therefore, there are [tex]\(10\)[/tex] choices for each of the 4 positions.
- Total number of four-digit codes:
[tex]\[
10 \times 10 \times 10 \times 10 = 10^4 = 10000
\][/tex]
2. Calculating codes starting with 7:
- The first digit must be 7, so there is only 1 choice for the first digit.
- Each of the remaining three digits can be any number from 0 to 9.
- Thus, for the remaining three positions:
[tex]\[
10 \times 10 \times 10 = 10^3 = 1000
\][/tex]
3. Determining the probability:
- The probability of the code beginning with 7 is the number of successful outcomes (codes starting with 7) divided by the total number of possible four-digit codes.
- The probability is:
[tex]\[
\frac{\text{Number of codes starting with 7}}{\text{Total number of 4-digit codes}} = \frac{1000}{10000} = 0.1
\][/tex]
4. Matching to the given choices:
- The expression that represents this probability correctly is:
[tex]\[
\frac{\left(10 P_1\right)\left({ }_9 P_3\right)}{10 P_4}
\][/tex]
Let's interpret this expression:
- [tex]\(10 P_1\)[/tex] represents the number of choices for the first digit (which includes 7).
- [tex]\(9 P_3\)[/tex] represents the permutation of the remaining 3 digits out of the 9 possibilities (since each can be a digit from 0 to 9).
- [tex]\(10 P_4\)[/tex] represents the permutation of 4 digits out of the 10 possibilities.
### Conclusion
The correct expression that determines the probability of the alarm code beginning with a 7 is:
[tex]\[
\frac{\left(10 P_1\right)\left({ }_9 P_3\right)}{10 P_4}
\][/tex]
