To determine the probability of a four-digit alarm code (using digits 0-9 without repetition) starting with a number greater than 7, we need to analyze the problem step-by-step:
1. Choice for the First Digit:
The code must begin with a digit greater than 7. The digits greater than 7 are 8 and 9. There are [tex]\(2\)[/tex] choices for the first digit.
2. Remaining Digits:
After selecting the first digit (from 8 or 9), we need to select and arrange the remaining three digits from the remaining 9 digits (since one digit has already been used). The order matters because we are creating a specific code, so we are dealing with permutations.
3. Calculating Permutations:
- Permutations for choosing 1 out of the 2 digits for the first place: [tex]\(\perm(2, 1)\)[/tex].
- Permutations for arranging the remaining 9 digits into the 3 remaining slots: [tex]\(\perm(9, 3)\)[/tex].
4. Total Permutations for the Four-Digit Code:
The total number of permutations for creating any four-digit code from the 10 digits without repetition is given by [tex]\(\perm(10, 4)\)[/tex].
5. Probability Expression:
To find the probability, we need the ratio of the favorable outcomes (codes starting with a digit greater than 7) to the total possible outcomes.
Now, putting these steps together, the favorable outcome can be expressed as:
[tex]\[
\frac{(\perm(2, 1) \cdot \perm(9, 3))}{\perm(10, 4)}
\][/tex]
This matches the given expression:
[tex]\[
\frac{\left({}_2P_1\right)\left(9P_3\right)}{10P_4}
\][/tex]
Therefore, the correct expression to determine the probability of the alarm code beginning with a number greater than 7 is:
[tex]\[
\boxed{\frac{\left({}_2P_1\right)\left(9P_3\right)}{10P_4}}
\][/tex]
