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

Options

A. 130
B. 131
C. 156
D. 158

Explanation

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

### Step-by-Step Solution:

1. **Count Three-Digit Numbers:**
- A three-digit number cannot start with 0. Therefore, the first digit (hundreds place) can be any of the digits 1, 2, 3, 4, or 5. This gives us 5 options.
- After choosing the first digit, we have 5 remaining digits (including 0) for the second digit (tens place).
- After choosing the first and second digits, we have 4 remaining digits for the third digit (units place).
- Therefore, the total number of three-digit numbers is calculated as follows:
\[ \text{Total three-digit numbers} = 5 \times 5 \times 4 = 100 \]

2. **Count Two-Digit Numbers:**
- A two-digit number also cannot start with 0. Therefore, the first digit (tens place) can be any of the digits 1, 2, 3, 4, or 5. This gives us 5 options.
- After choosing the first digit, we have 5 remaining digits (including 0) for the second digit (units place).
- Therefore, the total number of two-digit numbers is calculated as follows:
\[ \text{Total two-digit numbers} = 5 \times 5 = 25 \]

3. **Count One-Digit Numbers:**
- One-digit numbers can be any of the digits 1, 2, 3, 4, 5, or 0. This gives us 6 options.
- Therefore, the total number of one-digit numbers is:
\[ \text{Total one-digit numbers} = 6 \]

4. **Calculate the Total:**
- Now, we add the total numbers from each case:
\[ \text{Total numbers less than 1000} = \text{Total three-digit numbers} + \text{Total two-digit numbers} + \text{Total one-digit numbers} \]
\[ \text{Total numbers less than 1000} = 100 + 25 + 6 = 131 \]

### Final Answer:
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**.