A security alarm requires a four-digit code. The code can use the digits 0-9 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]$\frac{\left(2 P_1\right)\left(9 P_3\right)}{10 P_4}$[/tex]
[tex]$\frac{\left(2 C_1 M C_3\right)}{{ }_{10} C_4}$[/tex]
[tex]$\frac{\left(10 P_1\right)\left({ }_9 P_3\right)}{10 P_4}$[/tex]
[tex]$\frac{\left(10 C_1\right)\left(C_3\right)}{{ }_{10} C_4}$[/tex]
Answer (2)
Calculate the total number of possible four-digit codes without repetition using permutations: 10 P 4 = 5040 .
Determine the number of four-digit codes that begin with a number greater than 7: ( 2 P 1 ) ( 9 P 3 ) = 2 × 9 × 8 × 7 = 1008 .
Calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes: 10 P 4 ( 2 P 1 ) ( 9 P 3 ) .
The expression to determine the probability is: 10 P 4 ( 2 P 1 ) ( 9 P 3 ) .
Explanation
Analyze the problem Let's analyze the problem. We need to find the probability that a four-digit code (with no repeated digits from 0-9) begins with a number greater than 7. This means the first digit can be either 8 or 9. We need to determine the expression that represents this probability.
Calculate total possible codes First, let's find the total number of possible four-digit codes without repetition. This is a permutation problem because the order of the digits matters. We have 10 digits (0-9) and we want to arrange 4 of them. The total number of permutations is given by: 10 P 4 = ( 10 − 4 )! 10 ! = 6 ! 10 ! = 10 × 9 × 8 × 7 = 5040
Calculate codes starting with 8 or 9 Next, let's find the number of four-digit codes that begin with a number greater than 7. The possible digits are 8 and 9, so there are 2 choices for the first digit. After choosing the first digit, we have 9 remaining digits to choose from for the second digit, 8 for the third, and 7 for the fourth. This can be represented as: 2 × 9 × 8 × 7 We can also express the number of ways to arrange the remaining 3 digits from the remaining 9 as a permutation: 9 P 3 = ( 9 − 3 )! 9 ! = 6 ! 9 ! = 9 × 8 × 7 = 504 . So the number of such codes is 2 × 504 = 1008 . This can be written as ( 2 P 1 ) ( 9 P 3 ) since we are choosing 1 digit from 2 (8 or 9) for the first position and arranging 3 digits out of the remaining 9 for the other positions.
Calculate the probability The probability of the alarm code beginning with a number greater than 7 is the number of favorable outcomes (codes starting with 8 or 9) divided by the total number of possible outcomes (all possible four-digit codes): 10 P 4 ( 2 P 1 ) ( 9 P 3 ) = 10 × 9 × 8 × 7 2 × 9 × 8 × 7 = 5040 1008 = 5 1 = 0.2
Identify the correct expression Comparing the derived expression with the given options, we find that the correct expression is: 10 P 4 ( 2 P 1 ) ( 9 P 3 )
State the final answer Therefore, the expression that can be used to determine the probability of the alarm code beginning with a number greater than 7 is 10 P 4 ( 2 P 1 ) ( 9 P 3 ) .
Examples
Imagine you're designing a combination lock for a bike. You want to know the chances that someone could guess the combination if it starts with a specific number. This probability calculation helps you understand how secure the lock is based on the possible combinations and the likelihood of someone guessing the first number correctly.
To calculate the probability that a four-digit alarm code starts with a number greater than 7, we find the total number of possible codes and the codes that can start with either 8 or 9. This gives us the expression: 10 P 4 ( 2 P 1 ) ( 9 P 3 ) as the correct option. The final calculated probability is \frac{1}{5}.
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