Math Problem Statement
4 number code. No sequential numbers in either direction. No same number twice. Use numbers from 0 to 9. How many unique options does that leave?
Solution
We need to determine how many unique 4-digit codes can be formed under the following constraints:
- The numbers range from 0 to 9.
- No digit can be repeated.
- No sequential numbers in either direction (e.g., 12 or 21 is not allowed).
Step 1: Calculate the total number of codes without restrictions
If there were no restrictions except uniqueness, we would have:
- 10 choices for the first digit.
- 9 choices for the second digit.
- 8 choices for the third digit.
- 7 choices for the fourth digit.
This results in:
Step 2: Remove codes with sequential numbers
To account for the "no sequential numbers in either direction" restriction, we will determine how many codes violate this rule.
Counting invalid codes:
- Choosing the first digit: We have 10 choices.
- Choosing the second digit: We have 8 choices because we must exclude the digit that is sequentially adjacent to the first digit.
- Choosing the third digit: We have 7 choices (excluding the previous digit and any sequential digit).
- Choosing the fourth digit: We have 6 choices (excluding the previous digit and any sequential digit).
Thus, the number of valid codes is:
So, the final answer is 3,360 unique codes.
Would you like a breakdown with an example? Here are some related questions:
- What if the code length was 5 instead of 4?
- What if sequential numbers were allowed?
- What if repeated digits were allowed?
- What if we only used digits 1 to 9, excluding 0?
- Can we derive a general formula for different code lengths?
Tip: When applying constraints like "no adjacent digits," always count how many choices remain for each step to avoid overcounting.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Combinatorics
Permutations
Counting Principles
Formulas
Total permutations without restrictions: P(n, k) = n! / (n - k)!
Valid permutations considering no sequential numbers: Adjusted factorial choices per step
Theorems
Fundamental Principle of Counting
Suitable Grade Level
Grades 9-12
Related Recommendation
Unique Student Number Combinations with Non-Repeating Digits and Letters
How to Calculate the Number of 4-Digit Codes with Digits 0-3
Permutations of Digits 1,2,3,4,5 with Combined Last Two Digits
Generating 4-Digit Number Sequences from 1, 2, 3, 4, 5 by Combining Middle Digits
Finding the Number of Four-Digit Numbers from {0, 1, 1, 2, 3, 4}