Find the total number of non-negative integers less than 1000 for which the sum of the digits is 10

Options

A. A) 63
B. B) 66
C. C) 67
D. B) 56

Answer

To solve the problem of finding the total number of non-negative integers less than 1000 for which the sum of the digits is 10, we can break down the solution step by step.

### Step 1: Define the variables
We denote the three digits of a number as \(X\), \(Y\), and \(Z\). The requirement is that the sum of these digits must equal 10:
\[ X + Y + Z = 10 \]
where \(X\), \(Y\), and \(Z\) are the digits of the number.

### Step 2: Set up the equation for non-negative integer solutions
We need to find the non-negative integer solutions to the equation \(X + Y + Z = 10\). This can be solved using the "stars and bars" theorem, which states that the number of non-negative integer solutions to the equation \(x_1 + x_2 + ... + x_r = n\) is given by:
\[ \binom{n + r - 1}{r - 1} \]
In our case, \(n = 10\) (the sum) and \(r = 3\) (the number of digits).

### Step 3: Apply the formula
Substituting the values into the formula, we have:
\[ \text{Number of solutions} = \binom{10 + 3 - 1}{3 - 1} = \binom{12}{2} \]

### Step 4: Calculate \(\binom{12}{2}\)
Now we calculate \(\binom{12}{2}\):
\[ \binom{12}{2} = \frac{12 \times 11}{2 \times 1} = \frac{132}{2} = 66 \]
So, there are 66 non-negative integer solutions to the equation \(X + Y + Z = 10\).

### Step 5: Exclude invalid cases
Next, we need to exclude cases where any of the digits \(X\), \(Y\), or \(Z\) are greater than 9, as they cannot be valid single-digit numbers. The only way this can happen is if one of the digits is equal to 10.

The possible cases to exclude are:
1. \(X = 10, Y = 0, Z = 0\)
2. \(X = 0, Y = 10, Z = 0\)
3. \(X = 0, Y = 0, Z = 10\)

These three cases are invalid because they exceed the digit limit of 9.

### Step 6: Calculate the valid solutions
Thus, we subtract these invalid cases from the total:
\[ \text{Valid solutions} = 66 - 3 = 63 \]

### Final Answer
Therefore, the total number of non-negative integers less than 1000 for which the sum of the digits is 10 is:
\[ \boxed{63} \]