Math Problem Statement

create all possible three-digit combinations from the digits 1 , 2 , 3 , 4 , 5 , 1,2,3,4,5, and 6, where the structure is two-digit number + single-digit number, combine each two-digit pair with a remaining unused single-digit....example...12,3. 32, 1. 6,5 4. 45, 6. 32,4... Each combination will use each digit to compile all possibilities of sequences and combinations

Solution

To create all possible combinations as per the structure you've described, we start by forming all two-digit pairs from the digits 1,2,3,4,5,61, 2, 3, 4, 5, 6. Each combination will then use a remaining unused single-digit as the final digit in the format two-digit number,single-digit number\text{two-digit number}, \text{single-digit number}.

Here's a structured way to proceed:

  1. Generate two-digit pairs: We create every possible unique two-digit combination from the digits 11 through 66 (e.g., 1212, 1313, 1414, etc.), ensuring no digits are repeated within each two-digit pair.

  2. Assign an unused single-digit: For each two-digit pair, we select a single digit from the remaining numbers (excluding the digits used in the pair) to complete the combination.

  3. List all possibilities: We compile every combination in the form two-digit number,remaining digit\text{two-digit number}, \text{remaining digit}, ensuring each digit is used only once in each combination.

Let’s calculate the combinations according to these rules.I've created the combinations according to your specified structure. There are a total of 120 unique combinations. Here are the first few examples:

  1. 12,312,3
  2. 12,412,4
  3. 12,512,5
  4. 12,612,6
  5. 13,213,2
  6. 13,413,4
  7. 13,513,5
  8. 13,613,6
  9. 14,214,2
  10. 14,314,3

Would you like the complete list or have any questions on the methodology?

Here are some follow-up questions you might find useful:

  1. Would you like to explore combinations with repeated digits in the two-digit pair?
  2. Do you need combinations that involve a different set of digits or a different number structure?
  3. Would you like to see the combinations visualized in any particular way?
  4. Should we sort the results in a specific order, such as numerically or by unique two-digit pairs?
  5. Are there any additional constraints you'd like to add to these combinations?

Tip: Using permutations for generating combinations can be flexible; changing parameters (e.g., digit counts) will modify the pattern easily.

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Math Problem Analysis

Mathematical Concepts

Combinatorics
Permutations
Number Theory

Formulas

nPr = n! / (n - r)! for permutations of n objects taken r at a time

Theorems

Permutation principles for non-repeating combinations

Suitable Grade Level

Grades 8-10