The number of numbers less than 1000 that can be formed out of the digits 0,1,2,3,4 and 5, no digit being repeated, is:

Options

A. a. 111
B. b. 212
C. c. can't be determined
D. d. none of these

Answer

To find the total number of numbers less than 1000 that can be formed using the digits 0, 1, 2, 3, 4, and 5 without repetition, we will consider three cases: three-digit numbers, two-digit numbers, and one-digit numbers.

### Step 1: Count Three-Digit Numbers
1. **First Digit**: The first digit cannot be 0 (as it would not be a three-digit number). Therefore, we can choose from the digits 1, 2, 3, 4, or 5. This gives us 5 options.
2. **Second Digit**: After choosing the first digit, we have 5 digits left (including 0). Therefore, we can choose the second digit in 5 ways.
3. **Third Digit**: After choosing the first and second digits, we have 4 digits left. Thus, we can choose the third digit in 4 ways.

The total number of three-digit numbers can be calculated as:
\[ \text{Total Three-Digit Numbers} = 5 \times 5 \times 4 = 100 \]

### Step 2: Count Two-Digit Numbers
1. **First Digit**: The first digit cannot be 0, so we can again choose from the digits 1, 2, 3, 4, or 5. This gives us 5 options.
2. **Second Digit**: After choosing the first digit, we can choose the second digit from the remaining 5 digits (including 0). Thus, we have 5 options for the second digit.

The total number of two-digit numbers can be calculated as:
\[ \text{Total Two-Digit Numbers} = 5 \times 5 = 25 \]

### Step 3: Count One-Digit Numbers
For one-digit numbers, we can use any of the digits 0, 1, 2, 3, 4, or 5. Thus, we have:
\[ \text{Total One-Digit Numbers} = 6 \]

### Step 4: Calculate Total Numbers
Now, we can add the total numbers from all three cases:
\[ \text{Total Numbers} = \text{Total Three-Digit Numbers} + \text{Total Two-Digit Numbers} + \text{Total One-Digit Numbers} \]
\[ \text{Total Numbers} = 100 + 25 + 6 = 131 \]

### Final Answer
Thus, the total number of numbers less than 1000 that can be formed out of the digits 0, 1, 2, 3, 4, and 5, with no digit being repeated, is **131**.

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