Question

The default passcode on a cell phone is usually 4 digits, each 0-9. a. How many different passcodes are possible? b. If you can enter a 4-digit passcode in one second, about how long would it take you to try all possible passcodes?

Short answer

a. 10,000 passcodes b. Approximately 2 hours and 47 minutes

Step-by-step solution

Understand the Problem

Each digit in a 4-digit passcode ranges from 0 to 9. Therefore, we need to calculate how many different combinations of 4 digits we can form.

Calculate Total Combinations

Since each digit has 10 possible values (0 through 9), and there are 4 digits, we use the formula for calculating permutations with repetition: \[ 10 \times 10 \times 10 \times 10 = 10^4 \]This means there are 10,000 possible different passcodes.

Calculate Time to Try All Combinations

Each passcode takes 1 second to try. Given there are 10,000 passcodes, the total time required is simply: \[ 10,000 \text{ seconds} \].

Convert Time to Larger Units

To make the time more understandable, we convert seconds to hours: - There are 60 seconds in a minute, so it takes \( \frac{10,000}{60} \approx 166.67 \text{ minutes} \)- There are 60 minutes in an hour, thus it takes \( \frac{166.67}{60} \approx 2.78 \text{ hours} \)

Round Time Estimate

Finally, round 2.78 hours to a more practical figure, yielding approximately 2 hours and 47 minutes.

Key concepts

passcode combinations

When we talk about passcode combinations, we're looking at different sequences of numbers that one might use to unlock something like a phone. For a typical 4-digit passcode, each digit can be any number from 0 to 9. This is known as 'permutations with repetition,' meaning each position in the passcode can repeat numbers or use the same number more than once.
To figure out how many combinations exist, you consider each position in the passcode independently:

• There are 10 choices (from 0 to 9) for the first digit.
• There are 10 choices for the second digit, regardless of what the first digit was.
• Similarly, the third and fourth digits also each have 10 choices.

This results in a total of: \[ 10 \times 10 \times 10 \times 10 = 10^4 \]Which totals to 10,000 different possible passcode combinations. So, if you're trying to guess a 4-digit passcode, you'd have quite a lot to work through!

time estimation

Estimating the time to try and guess all possible passcode combinations involves some basic math but can be quite enlightening. If you could enter one 4-digit passcode per second, then you'd use 1 second per attempt.
Given there are 10,000 combinations in total (as calculated previously), it would take exactly 10,000 seconds to try them all.
However, seconds alone don't give us a sense of the actual time spent, so let's translate this into larger units of time.

• Converting 10,000 seconds into minutes: \( \frac{10,000}{60} \approx 166.67 \) minutes.
• Next, convert minutes into hours: \( \frac{166.67}{60} \approx 2.78 \) hours.

This means it would take you approximately 2.78 hours of continuous attempts to try every possible combination, highlighting how quickly time can add up when dealing with simple numerical sequences.

4-digit combinations

4-digit combinations are not only important for understanding passcodes but also serve as a fundamental example of permutations with repetition in mathematics. They explore how ordering elements matters, and how each element is independently selected.
Computing 4-digit permutations with repetition includes calculating, for each of the 4 positions, 10 different possibilities. It gives exactly 10,000 potential sequences.
This principle remains relevant in many fields. Whether it’s statistical modeling, simple coding, or even creating secure passwords. The vast number of combinations (10,000) illustrates the complexity that can arise from just a few variables, emphasizing the power and combinatorial nature of even basic digital sequences.

• Each digit’s choice is crucial, contributing to the total count.
• Recognizing how to compute permutations is foundational knowledge.

Mastery of these concepts can enhance problem-solving skills across various scenarios.