Get the answers you need at Westonci.ca, where our expert community is dedicated to providing you with accurate information. Discover reliable solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
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]
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]
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.