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:

  1. The numbers range from 0 to 9.
  2. No digit can be repeated.
  3. 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:

10×9×8×7=504010 \times 9 \times 8 \times 7 = 5040

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:

  1. Choosing the first digit: We have 10 choices.
  2. Choosing the second digit: We have 8 choices because we must exclude the digit that is sequentially adjacent to the first digit.
  3. Choosing the third digit: We have 7 choices (excluding the previous digit and any sequential digit).
  4. Choosing the fourth digit: We have 6 choices (excluding the previous digit and any sequential digit).

Thus, the number of valid codes is:

10×8×7×6=336010 \times 8 \times 7 \times 6 = 3360

So, the final answer is 3,360 unique codes.


Would you like a breakdown with an example? Here are some related questions:

  1. What if the code length was 5 instead of 4?
  2. What if sequential numbers were allowed?
  3. What if repeated digits were allowed?
  4. What if we only used digits 1 to 9, excluding 0?
  5. 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.

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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