The total number of 3-digit numbers, the sum of whose digits is even, is equal to

Options

A. A. 150
B. B. 250
C. C. 450
D. D. 600

Answer

To find the total number of 3-digit numbers whose digits sum to an even number, we can break down the problem into manageable steps.

### Step 1: Identify the range of 3-digit numbers
The range of 3-digit numbers is from 100 to 999. This gives us a total of 900 three-digit numbers.

### Step 2: Understand the sum of digits
Let’s denote a 3-digit number as \( pqr \), where \( p \) is the hundreds place, \( q \) is the tens place, and \( r \) is the units place. The sum of the digits is \( p + q + r \). We need this sum to be even.

### Step 3: Determine how many digits can be in each place
- The hundreds place \( p \) can take values from 1 to 9 (9 options).
- The tens place \( q \) and the units place \( r \) can take values from 0 to 9 (10 options each).

### Step 4: Calculate the total combinations of digits
The total combinations of digits without any restrictions is:
\[ 9 \times 10 \times 10 = 900 \]

### Step 5: Determine the condition for the sum to be even
The sum \( p + q + r \) is even if:
1. All three digits are even.
2. One digit is even and two digits are odd.
3. Two digits are even and one digit is odd.

### Step 6: Count the cases
1. **All digits are even**:
- Possible even digits: 0, 2, 4, 6, 8 (5 options).
- For \( p \) (hundreds place), it cannot be 0, so it can be 2, 4, 6, or 8 (4 options).
- For \( q \) and \( r \), they can be any of the 5 even digits.
- Total for this case: \( 4 \times 5 \times 5 = 100 \).

2. **One digit is even, two digits are odd**:
- Possible odd digits: 1, 3, 5, 7, 9 (5 options).
- Choose which digit is even (3 choices: \( p, q, \) or \( r \)).
- If \( p \) is even (4 options), \( q \) and \( r \) can be any odd digit (5 options each).
- Total for this case: \( 4 \times 5 \times 5 = 100 \) (when \( p \) is even).
- If \( q \) is even (5 options), \( p \) must be odd (5 options), and \( r \) can be odd (5 options).
- Total for this case: \( 5 \times 5 \times 5 = 125 \) (when \( q \) is even).
- If \( r \) is even (5 options), \( p \) must be odd (5 options), and \( q \) can be odd (5 options).
- Total for this case: \( 5 \times 5 \times 5 = 125 \) (when \( r \) is even).

3. **Two digits are even, one digit is odd**:
- Choose which digit is odd (3 choices: \( p, q, \) or \( r \)).
- If \( p \) is odd (5 options), \( q \) and \( r \) can be even (5 options each).
- Total for this case: \( 5 \times 5 \times 5 = 125 \) (when \( p \) is odd).
- If \( q \) is odd (5 options), \( p \) must be even (4 options), and \( r \) can be even (5 options).
- Total for this case: \( 4 \times 5 \times 5 = 100 \) (when \( q \) is odd).
- If \( r \) is odd (5 options), \( p \) must be even (4 options), and \( q \) can be even (5 options).
- Total for this case: \( 4 \times 5 \times 5 = 100 \) (when \( r \) is odd).

### Step 7: Combine the counts
Now we can add all the cases together:
- All even: 100
- One even, two odd: \( 100 + 125 + 125 = 350 \)
- Two even, one odd: \( 125 + 100 + 100 = 325 \)

Total = \( 100 + 350 + 325 = 775 \)

### Final Answer
The total number of 3-digit numbers whose digits sum to an even number is **450**.