A security alarm requires a four-digit code. The code can use the digits [tex]$0-9$[/tex] and the digits cannot be repeated. Which expression can be used to determine the probability of the alarm code beginning with a number greater than 7?
[tex]$\frac{\left({ }_2 P_1\right)\left({ }_5 P_3\right)}{10 P_4}$[/tex]
[tex]$\frac{\left({ }_2 C_1\right)\left({ }_9 C_3\right)}{{ }_{10} C_4}$[/tex]
[tex]$\frac{\left(10^f\right)^{(2 f 3)}}{10^{f 4}}$[/tex]
[tex]$\frac{\left(10^6-1\right)(5)}{10^6}$[/tex]
Answer (2)
Calculate the total number of possible four-digit codes without repetition: 10 P 4 = 5040 .
Calculate the number of four-digit codes that begin with a digit greater than 7: 2 P 1 × 9 P 3 = 2 × 504 = 1008 .
Calculate the probability as the ratio of favorable outcomes to total outcomes: 5040 1008 .
The expression for the probability is: 10 P 4 ( 2 P 1 ) ( 9 P 3 ) .
Explanation
Understand the problem We are asked to find the expression that represents the probability of a four-digit security code (with no repeated digits) starting with a digit greater than 7. The possible digits are 0 through 9.
Calculate the total number of possible codes First, let's find the total number of possible four-digit codes without repetition. Since we have 10 digits (0-9) and we need to choose 4 of them without repetition, the total number of possible codes is given by the permutation formula: 10 P 4 = ( 10 − 4 )! 10 ! = 6 ! 10 ! = 10 × 9 × 8 × 7 = 5040
Calculate the number of codes starting with a digit greater than 7 Next, let's find the number of four-digit codes that begin with a digit greater than 7. The digits greater than 7 are 8 and 9. So, the first digit can be either 8 or 9, which gives us 2 choices for the first digit. After choosing the first digit, we have 9 remaining digits to choose from for the second digit, 8 remaining digits for the third digit, and 7 remaining digits for the fourth digit. Therefore, the number of ways to choose the remaining 3 digits is given by the permutation formula: 9 P 3 = ( 9 − 3 )! 9 ! = 6 ! 9 ! = 9 × 8 × 7 = 504
Calculate the number of favorable outcomes The number of favorable outcomes (codes starting with a digit greater than 7) is the product of the number of choices for the first digit (2) and the number of ways to arrange the remaining 3 digits (504): 2 × 504 = 1008 This can also be written as 2 P 1 × 9 P 3 = 2 × ( 9 × 8 × 7 ) = 1008 .
Calculate the probability The probability of the alarm code beginning with a number greater than 7 is the ratio of the number of favorable outcomes to the total number of possible outcomes: Total number of possible outcomes Number of favorable outcomes = 5040 1008 = 10 × 9 × 8 × 7 2 × 9 × 8 × 7 = 10 2 = 5 1 = 0.2 This can be expressed as: 10 P 4 2 P 1 × 9 P 3
Identify the correct expression Comparing the calculated probability expression with the given options, we find that the correct expression is: 10 P 4 ( 2 P 1 ) ( 9 P 3 )
Examples
This type of probability calculation is useful in cryptography and security, where you need to assess the likelihood of an attacker guessing a password or code. For example, if a system uses a 6-character password with letters and numbers, you can calculate the probability of an attacker guessing the password within a certain number of attempts. This helps in determining the strength of the password and the security measures needed to protect the system.
The probability of a four-digit security code starting with a digit greater than 7 can be calculated using the expression 10 P 4 ( 2 P 1 ) ( 9 P 3 ) . This expression represents the number of favorable outcomes over the total outcomes. Therefore, the correct choice is the first expression provided.
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