Math Problem Statement

How do I calculate the number of combinations from something like a 3-digit code and 6 buttons with turn on or off.

Solution

To calculate the number of combinations for a situation where you have a 3-digit code and 6 buttons that can each be turned on or off, you need to consider both aspects separately and then combine the results.

Step 1: Calculate the number of combinations for the 3-digit code

  • A 3-digit code usually consists of digits from 0 to 9.
  • So, each digit has 10 possible choices.
  • For a 3-digit code, the number of possible combinations is:

10×10×10=103=100010 \times 10 \times 10 = 10^3 = 1000

Step 2: Calculate the number of combinations for the 6 buttons

  • Each button has 2 states: on or off.
  • Therefore, for 6 buttons, the number of possible combinations is:

2×2×2×2×2×2=26=642 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^6 = 64

Step 3: Combine the results

  • Since the 3-digit code and the 6-button configurations are independent of each other, the total number of combinations is the product of the two:

1000×64=640001000 \times 64 = 64000

So, there are 64,000 possible combinations in total.

Would you like more details, or do you have any other questions?

Here are 5 related questions you might be interested in:

  1. How would the calculation change if the 3-digit code allowed repeated digits?
  2. What if there were 8 buttons instead of 6?
  3. How can the number of combinations be affected if the buttons have more than two states (e.g., on, off, and standby)?
  4. What is the number of combinations if the code were 4 digits instead of 3?
  5. How would you calculate the number of combinations if there were restrictions, like specific digits or buttons that must be used?

Tip: When dealing with combinations, always ensure to check if there are any restrictions or conditions that might reduce the total count.

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Math Problem Analysis

Mathematical Concepts

Combinatorics
Basic Arithmetic

Formulas

Number of combinations for a 3-digit code: 10^3
Number of combinations for 6 buttons: 2^6

Theorems

Multiplication Principle in Combinatorics

Suitable Grade Level

Grades 7-9