To find the total number of 3-digit numbers whose sum of digits is 10, we will break down the problem step by step.
### Step 1: Define the digits
Let the three digits of the 3-digit number be represented as \(a\), \(b\), and \(c\), where:
- \(a\) is the hundreds place (1 to 9, since it cannot be zero),
- \(b\) is the tens place (0 to 9),
- \(c\) is the units place (0 to 9).
We need to satisfy the equation:
\[ a + b + c = 10 \]
### Step 2: Determine the range for \(a\)
Since \(a\) cannot be zero, the possible values for \(a\) are from 1 to 9. We will consider different cases based on the value of \(a\).
### Step 3: Case Analysis
We will analyze cases for each possible value of \(a\) from 1 to 9.
- **Case 1**: \(a = 1\)
\[ b + c = 10 - 1 = 9 \]
Possible pairs \((b, c)\) are: (0,9), (1,8), (2,7), (3,6), (4,5), (5,4), (6,3), (7,2), (8,1), (9,0) → 10 pairs.
- **Case 2**: \(a = 2\)
\[ b + c = 10 - 2 = 8 \]
Possible pairs: (0,8), (1,7), (2,6), (3,5), (4,4), (5,3), (6,2), (7,1), (8,0) → 9 pairs.
- **Case 3**: \(a = 3\)
\[ b + c = 10 - 3 = 7 \]
Possible pairs: (0,7), (1,6), (2,5), (3,4), (4,3), (5,2), (6,1), (7,0) → 8 pairs.
- **Case 4**: \(a = 4\)
\[ b + c = 10 - 4 = 6 \]
Possible pairs: (0,6), (1,5), (2,4), (3,3), (4,2), (5,1), (6,0) → 7 pairs.
- **Case 5**: \(a = 5\)
\[ b + c = 10 - 5 = 5 \]
Possible pairs: (0,5), (1,4), (2,3), (3,2), (4,1), (5,0) → 6 pairs.
- **Case 6**: \(a = 6\)
\[ b + c = 10 - 6 = 4 \]
Possible pairs: (0,4), (1,3), (2,2), (3,1), (4,0) → 5 pairs.
- **Case 7**: \(a = 7\)
\[ b + c = 10 - 7 = 3 \]
Possible pairs: (0,3), (1,2), (2,1), (3,0) → 4 pairs.
- **Case 8**: \(a = 8\)
\[ b + c = 10 - 8 = 2 \]
Possible pairs: (0,2), (1,1), (2,0) → 3 pairs.
- **Case 9**: \(a = 9\)
\[ b + c = 10 - 9 = 1 \]
Possible pairs: (0,1), (1,0) → 2 pairs.
### Step 4: Total the pairs
Now, we will sum the number of pairs from all cases:
- Case 1: 10 pairs
- Case 2: 9 pairs
- Case 3: 8 pairs
- Case 4: 7 pairs
- Case 5: 6 pairs
- Case 6: 5 pairs
- Case 7: 4 pairs
- Case 8: 3 pairs
- Case 9: 2 pairs
Total pairs = \(10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 = 54\).
### Final Answer
Thus, the total number of 3-digit numbers whose sum of digits is 10 is **54**.
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