Answer :
Let's break down how we can determine the correct expression for the probability of the alarm code beginning with a number greater than 7, given that the code is a four-digit number using the digits 0-9 without repetition.
### Step-by-Step Solution
1. Digits Available and First Digit Choice:
- We have the digits 0 through 9, so there are 10 digits available.
- The first digit must be greater than 7. The digits greater than 7 are 8 and 9.
- Thus, there are 2 choices for the first digit.
2. Remaining Digits:
- After choosing the first digit, we are left with 9 digits.
- We need to select and arrange 3 more digits for the remaining positions.
3. Permutations for Remaining Digits:
- The number of ways to select and arrange 3 digits out of the remaining 9 digits (after choosing the first digit) is given by permutations.
- The number of permutations of 3 digits out of 9 is [tex]\( P(9, 3) \)[/tex].
4. Total Permutations for a 4-Digit Code:
- The total number of permutations for selecting and arranging any 4 digits out of 10 is [tex]\( P(10, 4) \)[/tex].
### Forming the Probability Expression
- Probability Formula:
The probability that the alarm code starts with a digit greater than 7 is given by the ratio of the favorable outcomes (codes starting with 8 or 9) to the total outcomes (all possible four-digit codes).
[tex]\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \][/tex]
- Favorable Outcomes:
- We have 2 choices for the first digit.
- For each choice of the first digit, there are [tex]\( P(9, 3) \)[/tex] ways to arrange the remaining 3 digits.
Therefore, the number of favorable outcomes is given by:
[tex]\[ 2 \times P(9, 3) \][/tex]
- Total Outcomes:
The total number of ways to create any 4-digit code from the digits 0-9 without repetition is [tex]\( P(10, 4) \)[/tex].
Therefore, the correct expression to calculate the probability that the alarm code begins with a number greater than 7 is:
[tex]\[ \frac{2 \times P(9, 3)}{P(10, 4)} \][/tex]
### Matching to Given Choices
Among the provided options:
[tex]\[ \frac{(2 P_1)(9 P_3)}{10 P_4} \][/tex]
matches our derived probability expression, where:
- [tex]\( P_1 \)[/tex] indicates the number of choices for the first digit.
- [tex]\( 9 P_3 \)[/tex] indicates the permutations of the remaining 3 digits out of the 9 available digits.
- [tex]\( 10 P_4 \)[/tex] represents the total permutations for selecting and arranging 4 out of 10 digits.
Hence, the correct expression is:
[tex]\[ \frac{(2 P_1)(9 P_3)}{10 P_4} \][/tex]
The result, therefore, is:
[tex]\[ 1 \][/tex] (corresponding to the first option in the list).
### Step-by-Step Solution
1. Digits Available and First Digit Choice:
- We have the digits 0 through 9, so there are 10 digits available.
- The first digit must be greater than 7. The digits greater than 7 are 8 and 9.
- Thus, there are 2 choices for the first digit.
2. Remaining Digits:
- After choosing the first digit, we are left with 9 digits.
- We need to select and arrange 3 more digits for the remaining positions.
3. Permutations for Remaining Digits:
- The number of ways to select and arrange 3 digits out of the remaining 9 digits (after choosing the first digit) is given by permutations.
- The number of permutations of 3 digits out of 9 is [tex]\( P(9, 3) \)[/tex].
4. Total Permutations for a 4-Digit Code:
- The total number of permutations for selecting and arranging any 4 digits out of 10 is [tex]\( P(10, 4) \)[/tex].
### Forming the Probability Expression
- Probability Formula:
The probability that the alarm code starts with a digit greater than 7 is given by the ratio of the favorable outcomes (codes starting with 8 or 9) to the total outcomes (all possible four-digit codes).
[tex]\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \][/tex]
- Favorable Outcomes:
- We have 2 choices for the first digit.
- For each choice of the first digit, there are [tex]\( P(9, 3) \)[/tex] ways to arrange the remaining 3 digits.
Therefore, the number of favorable outcomes is given by:
[tex]\[ 2 \times P(9, 3) \][/tex]
- Total Outcomes:
The total number of ways to create any 4-digit code from the digits 0-9 without repetition is [tex]\( P(10, 4) \)[/tex].
Therefore, the correct expression to calculate the probability that the alarm code begins with a number greater than 7 is:
[tex]\[ \frac{2 \times P(9, 3)}{P(10, 4)} \][/tex]
### Matching to Given Choices
Among the provided options:
[tex]\[ \frac{(2 P_1)(9 P_3)}{10 P_4} \][/tex]
matches our derived probability expression, where:
- [tex]\( P_1 \)[/tex] indicates the number of choices for the first digit.
- [tex]\( 9 P_3 \)[/tex] indicates the permutations of the remaining 3 digits out of the 9 available digits.
- [tex]\( 10 P_4 \)[/tex] represents the total permutations for selecting and arranging 4 out of 10 digits.
Hence, the correct expression is:
[tex]\[ \frac{(2 P_1)(9 P_3)}{10 P_4} \][/tex]
The result, therefore, is:
[tex]\[ 1 \][/tex] (corresponding to the first option in the list).
