The number of numbers, lying between 99 and 1000 that can be made from the digits 2, 3, 7, 0, 8 and 6 when the digits occur only once in each number, is

Options

A. 100
B. 90
C. 120
D. 80

Answer

To find the number of three-digit numbers that can be formed using the digits 2, 3, 7, 0, 8, and 6, where each digit can only be used once, we need to follow these steps:

### Step 1: Identify the Range
We are looking for three-digit numbers that lie between 100 and 999. This means we need to consider numbers with three digits only.

**Hint:** Remember that a three-digit number cannot start with 0.

### Step 2: Determine the First Digit
The first digit of a three-digit number cannot be 0. Therefore, the possible choices for the first digit are 2, 3, 7, 8, and 6. This gives us 5 options.

**Hint:** List out the digits and eliminate 0 for the first position.

### Step 3: Choose the Second Digit
Once the first digit is chosen, we have 5 remaining digits (including 0). We can choose any of these 5 digits for the second position.

**Hint:** Remember that you can use 0 in the second position.

### Step 4: Choose the Third Digit
After selecting the first and second digits, we will have 4 digits left to choose from for the third position.

**Hint:** Ensure that the third digit is chosen from the remaining digits after the first two have been selected.

### Step 5: Calculate the Total Combinations
Now, we can calculate the total number of three-digit numbers that can be formed:
- Choices for the first digit: 5
- Choices for the second digit: 5 (after choosing the first digit)
- Choices for the third digit: 4 (after choosing the first and second digits)

Thus, the total number of three-digit numbers is:
\[ \text{Total Numbers} = 5 \times 5 \times 4 \]

### Step 6: Perform the Calculation
Calculating the above expression:
\[ 5 \times 5 = 25 \]
\[ 25 \times 4 = 100 \]

### Final Answer
The total number of three-digit numbers that can be formed from the digits 2, 3, 7, 0, 8, and 6, where each digit is used only once, is **100**.

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