How many three-digit numbers are there that have no repeating digits, do not contain the number zero, and all three digits in the number add up to 10?  use equations, show all your work and use organized counting strategies

We are tasked with determining how many three-digit numbers exist that:

1. Have no repeating digits.
2. Do not contain the digit 0.
3. The sum of their digits equals 10.

### Step 1: Represent the digits of the three-digit number
Let the digits of the number be represented as \( d_1, d_2, d_3 \), where:

- \( d_1 \) is the hundreds digit.
- \( d_2 \) is the tens digit.
- \( d_3 \) is the units digit.

Thus, we have:
\[
d_1 + d_2 + d_3 = 10
\]
Additionally, \( d_1, d_2, d_3 \) are digits between 1 and 9 (since 0 is excluded) and they must be distinct.

### Step 2: List the possible combinations
We need to find sets of three distinct digits from the set \( \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \) such that their sum equals 10.

Let's now enumerate all possible sets of distinct digits that satisfy this condition.

#### Finding combinations:
- Consider the equation \( d_1 + d_2 + d_3 = 10 \). We will try to find all combinations of \( d_1, d_2, d_3 \) from the set \( \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \) without repeating digits.
- By organized counting, we systematically check for all distinct digits whose sum is 10.

The possible combinations of digits that sum to 10 are:
- \( \{1, 2, 7\} \)
- \( \{1, 3, 6\} \)
- \( \{1, 4, 5\} \)
- \( \{2, 3, 5\} \)

These are the only distinct combinations of three digits from \( \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \) that sum to 10.

### Step 3: Permutations of each combination
Each combination can form different three-digit numbers by permuting the digits. For example, for the combination \( \{1, 2, 7\} \), we can have the following numbers:
- 127, 172, 217, 271, 712, 721

The number of distinct permutations for each set of digits is \( 3! = 6 \).

Thus, for each of the four combinations, we have 6 distinct numbers.

### Step 4: Total number of numbers
Since there are 4 distinct combinations of digits, and each combination results in 6 different numbers, the total number of three-digit numbers is:
\[
4 \times 6 = 24
\]

### Final Answer:
There are \( \boxed{24} \) three-digit numbers that satisfy the given conditions.

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Would you like more details, or do you have any questions?  
Here are some related questions you might find interesting:

1. How would the answer change if the digit 0 was allowed?
2. What if the digits could repeat, but still summed to 10?
3. What if the sum of the digits had to be 15 instead of 10?
4. How many three-digit numbers have distinct digits that are prime?
5. How many four-digit numbers have no repeating digits and sum to 15?

**Tip**: Organized counting (listing possibilities systematically) helps avoid missing any valid combinations!

How many three-digit numbers are there that have no repeating digits, do not contain the number zero, and all three digits in the number add up to 10? Use equations, show all your work, and use organized counting strategies.

Discover how to find the number of three-digit numbers with distinct digits, excluding zero, where the sum of the digits equals 10. This problem demonstrates organized counting strategies and permutations, suitable for Grades 9-12.

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