The number of +ve integers which can be formed by using any number of digits from 0 1 2 3 4 5 but using each digit not more that once in each number is

Options

A. 1200
B. 1500
C. 1600
D. 1630

Answer

To find the number of positive integers that can be formed using the digits 0, 1, 2, 3, 4, and 5, with each digit used at most once in each number, we can break down the problem into cases based on the number of digits in the integers.

### Step-by-Step Solution:

1. **Identify the Digits**: The available digits are 0, 1, 2, 3, 4, and 5. We have a total of 6 digits.

2. **Case 1: One-Digit Numbers**:
- The possible one-digit positive integers can only be formed using the digits 1, 2, 3, 4, and 5 (since 0 is not a positive integer).
- Thus, the number of one-digit numbers = 5.

3. **Case 2: Two-Digit Numbers**:
- The first digit cannot be 0 (to ensure it's a two-digit number). Therefore, we have 5 choices for the first digit (1, 2, 3, 4, or 5).
- After choosing the first digit, we can choose any of the remaining 5 digits (including 0) for the second digit.
- Thus, the number of two-digit numbers = 5 (choices for the first digit) × 5 (choices for the second digit) = 25.

4. **Case 3: Three-Digit Numbers**:
- The first digit again cannot be 0. So we have 5 choices for the first digit.
- After choosing the first digit, we have 5 remaining digits (including 0) for the second digit and 4 remaining digits for the third digit.
- Thus, the number of three-digit numbers = 5 (first digit) × 5 (second digit) × 4 (third digit) = 100.

5. **Case 4: Four-Digit Numbers**:
- The first digit cannot be 0, giving us 5 choices.
- After choosing the first digit, we have 5 remaining digits for the second digit, 4 for the third digit, and 3 for the fourth digit.
- Thus, the number of four-digit numbers = 5 × 5 × 4 × 3 = 300.

6. **Case 5: Five-Digit Numbers**:
- The first digit cannot be 0, giving us 5 choices.
- After choosing the first digit, we have 5 remaining digits for the second digit, 4 for the third digit, 3 for the fourth digit, and 2 for the fifth digit.
- Thus, the number of five-digit numbers = 5 × 5 × 4 × 3 × 2 = 600.

7. **Case 6: Six-Digit Numbers**:
- The first digit cannot be 0, giving us 5 choices.
- After choosing the first digit, we have 5 remaining digits for the second digit, 4 for the third digit, 3 for the fourth digit, 2 for the fifth digit, and 1 for the sixth digit.
- Thus, the number of six-digit numbers = 5 × 5 × 4 × 3 × 2 × 1 = 1200.

8. **Total Count**:
- Now, we add all the cases together:
- Total = Number of one-digit numbers + Number of two-digit numbers + Number of three-digit numbers + Number of four-digit numbers + Number of five-digit numbers + Number of six-digit numbers
- Total = 5 + 25 + 100 + 300 + 600 + 1200 = 2230.

### Final Answer:
The total number of positive integers that can be formed is **2230**.