How many 3 digit odd numbers can be formed with the digits 1,3,5,6,8?

Answer

To find the number of 3-digit odd numbers that can be formed using the digits 1, 3, 5, 6, and 8, we can follow these steps:

### Step 1: Identify the odd digits
The odd digits available from the given set are 1, 3, and 5. Since we want to form odd numbers, the unit's place (the last digit) must be one of these odd digits.

### Step 2: Choose the unit's place
We have 3 choices for the unit's place:
- It can be 1
- It can be 3
- It can be 5

### Step 3: Choose the hundreds and tens places
After selecting the unit's place, we need to fill the hundreds and tens places. The remaining digits after choosing one for the unit's place will be used for these two positions.

#### Case Analysis:
1. **If the unit's place is 1:**
- Remaining digits: 3, 5, 6, 8 (4 choices)
- Choose one for the hundreds place (4 options), then choose one for the tens place from the remaining 3 digits.
- Total combinations = 4 (choices for hundreds) × 3 (choices for tens) = 12.

2. **If the unit's place is 3:**
- Remaining digits: 1, 5, 6, 8 (4 choices)
- Total combinations = 4 (choices for hundreds) × 3 (choices for tens) = 12.

3. **If the unit's place is 5:**
- Remaining digits: 1, 3, 6, 8 (4 choices)
- Total combinations = 4 (choices for hundreds) × 3 (choices for tens) = 12.

### Step 4: Calculate the total
Now, we add the combinations from all three cases:
- Total = 12 (for unit's place 1) + 12 (for unit's place 3) + 12 (for unit's place 5) = 36.

### Final Answer
Thus, the total number of 3-digit odd numbers that can be formed is **36**.

---