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Sagot :
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]
### 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]
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