oseph found a cell phone and wants to return it to its owner. The phone requires a 4-digit code to gain access to the contact list. What is the probability that Joseph will be able to uniock the phone on is first attempt? (Assume that the numbers cannot be used more than one time.) - Does the order the numbers sie entered matter? - Will all of the digits \( (0,1,2,3,4,5,6,7,8,9) \) be used? - How many different ways can four digits be arranged? - P (Joseph chooses the correct code) \( = \)

To find the probability that Joseph can unlock the phone on his first attempt, let's break it down. Since the phone requires a 4-digit code and numbers cannot be repeated, we start with 10 available digits. 1. The first digit has 10 options (0-9). 2. The second digit has 9 options (one digit has been used). 3. The third digit has 8 options. 4. The fourth digit has 7 options. So, the total number of unique 4-digit combinations is calculated by multiplying the number of options for each position: \[ 10 \times 9 \times 8 \times 7 = 5040 \] This means there are 5040 different 4-digit codes possible. Now, if Joseph is guessing, he has only 1 correct code among these 5040 possibilities, which gives the probability of guessing correctly on the first attempt as: \[ P(\text{correct code}) = \frac{1}{5040} \] So, the probability that Joseph successfully unlocks the phone on his first try is a mere \( \frac{1}{5040} \). Good luck, Joseph!