How many 6-digit numbers have all their digits either all odd or all even?

Options

A. 31250
B. 28125
C. 15625
D. None of these

Answer

To solve the problem of how many 6-digit numbers have all their digits either all odd or all even, we can break it down into two separate cases: one for all odd digits and one for all even digits.

### Step 1: Count 6-digit numbers with all odd digits
1. **Identify the odd digits**: The odd digits available are 1, 3, 5, 7, and 9. This gives us a total of 5 odd digits.
2. **Determine the positions**: A 6-digit number has 6 positions (from the first digit to the sixth digit).
3. **Calculate the total combinations**: Since all 6 digits can be any of the 5 odd digits, the total number of combinations is calculated as:
\[ 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 5^6 = 15625 \]

### Step 2: Count 6-digit numbers with all even digits
1. **Identify the even digits**: The even digits available are 0, 2, 4, 6, and 8. This gives us a total of 5 even digits.
2. **Determine the first digit**: The first digit cannot be 0 (as it would make it a 5-digit number), so we have 4 options for the first digit (2, 4, 6, or 8).
3. **Determine the remaining digits**: The remaining 5 digits can be any of the 5 even digits (including 0).
4. **Calculate the total combinations**: The total number of combinations is calculated as:
\[ 4 \times 5 \times 5 \times 5 \times 5 \times 5 = 4 \times 5^5 = 4 \times 3125 = 12500 \]

### Step 3: Combine the results
1. **Total 6-digit numbers**: Now, we add the total numbers from both cases:
\[ 15625 \text{ (odd)} + 12500 \text{ (even)} = 28125 \]

### Final Answer
The total number of 6-digit numbers that have all their digits either all odd or all even is **28,125**.

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