To determine the probability that a four-digit code starts with a number greater than 7, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes.
### Step 1: Total Number of Possible 4-Digit Codes
Since the digits in the code range from 0 to 9 and cannot be repeated, we can use permutations to calculate the total number of 4-digit codes:
[tex]\[
_{10}P_{4} \text{(Permutations of 4 digits out of 10)}
\][/tex]
[tex]\[
= \frac{10!}{(10-4)!}
= \frac{10!}{6!}
= 10 \times 9 \times 8 \times 7
= 5040
\][/tex]
### Step 2: Number of Choices for the First Digit Greater than 7
Digits greater than 7 are 8 and 9, giving us:
[tex]\[
2 \text{ choices for the first digit}
\][/tex]
### Step 3: Number of Choices for the Remaining 3 Digits
Once the first digit is chosen, we have 9 digits left, and we need to choose 3 of them. The number of ways to choose 3 digits from 9 (without regard for order) is:
[tex]\[
_{9}P_{3} \text{(Permutations of 3 digits out of 9)}
\][/tex]
[tex]\[
= \frac{9!}{(9-3)!}
= \frac{9!}{6!}
= 9 \times 8 \times 7
= 504
\][/tex]
### Step 4: Total Number of Favorable 4-Digit Combinations
The number of favorable outcomes where the first digit is greater than 7 can be calculated by multiplying the number of choices for the first digit by the number of choices for the remaining digits:
[tex]\[
\text{Total favorable combinations} = 2 \times 504
\][/tex]
### Step 5: Probability Calculation
Finally, the probability is the ratio of the number of favorable outcomes to the total number of possible outcomes:
[tex]\[
\text{Probability} = \frac{\text{Number of favorable combinations}}{\text{Total number of possible combinations}}
\][/tex]
[tex]\[
= \frac{2 \times 504}{5040}
= 0.2
\][/tex]
So the expression that can be used to determine this probability is:
[tex]\[
\frac{\left({ }_2 P_1\right)\left({ }_9 P_3\right)}{10 P_4}
\][/tex]
Therefore, the correct option is:
[tex]\[
(-) \frac{\left({ }_2 P_1\right)\left({ }_9 P_3\right)}{10 P_4}
\][/tex]
