https://r2.erweima.ai/mathbot/image/1718582549219jgrviun9.png

To solve the problem of determining the number of possible three-digit pin codes and the probability of guessing the correct one on the first try, given an 8-key keypad with repetition allowed, we proceed as follows:

1. **Calculate the Number of Possible Codes:**

   Since the thief can choose any of the 8 digits (0-7) for each of the three positions in the pin code, the number of possible codes is calculated by:
   \[
   8 \times 8 \times 8 = 8^3
   \]

   Simplifying this:
   \[
   8^3 = 512
   \]

   So, the number of possible codes is 512.

2. **Determine the Probability of a Correct Guess:**

   The probability of guessing the correct code on the first try is the reciprocal of the number of possible codes since each code is equally likely:
   \[
   \text{Probability} = \frac{1}{512}
   \]

Therefore, the answers are:
- The number of possible codes is **512**.
- The probability of a correct guess on the first try is **\(\frac{1}{512}\)**.

Would you like further details or have any questions? Here are some more questions you might find interesting:

1. What is the probability of guessing a four-digit pin code on the first try using a 10-key keypad?
2. How many possible combinations are there for a five-character password using lowercase letters?
3. What is the probability of guessing a 6-character alphanumeric password on the first try?
4. If repetition is not allowed, how many different three-digit codes can be formed from an 8-key keypad?
5. What is the probability of guessing a two-digit pin code with a 5-key keypad?

**Tip:** Understanding basic principles of probability and combinatorics can significantly help in solving similar problems effectively.

Explore the calculation of the number of possible three-digit pin codes (512) and the probability of guessing the correct one on the first try using an 8-key keypad. This solution provides insights into combinatorics and probability theory. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation