how many positive integers less than 4000 with no repeated digits can be formed where each digit is either 2. 3. 4. 5 or 6?
2,304
To find the number of positive integers less than 4000 with no repeated digits using only the digits 2, 3, 4, 5, and 6, we can break down the problem into multiple cases.
1. For the thousands place, we can only use the digits 2, 3, 4, and 5 since integers less than 4000 cannot have 6 as the thousands digit. So there are 4 choices for the thousands place.
2. For the hundreds place, we can use any of the remaining 4 digits (excluding the digit already used in the thousands place). Since repetition is not allowed, there are 4 choices for the hundreds place.
3. Similarly, for the tens place, we have 3 choices remaining.
4. Finally, for the units place, we have 2 choices remaining.
Multiplying the number of choices together for each place, we get the total number of positive integers less than 4000 with no repeated digits as:
4 choices for thousands place × 4 choices for hundreds place × 3 choices for tens place × 2 choices for units place = 4 × 4 × 3 × 2 = 96.
Therefore, there are 96 positive integers less than 4000 with no repeated digits that can be formed using only the digits 2, 3, 4, 5, and 6.
To find the total number of positive integers with no repeated digits that can be formed using the digits 2, 3, 4, 5, and 6, we can follow these steps:
Step 1: Count the number of choices for each digit:
- The first digit can be any of the 5 given digits (2, 3, 4, 5, or 6). So, we have 5 choices for the first digit.
- The second digit can be any of the remaining 4 digits (since we can't repeat any digits). So, we have 4 choices for the second digit.
- The third digit can be any of the remaining 3 digits. So, we have 3 choices for the third digit.
- The fourth digit can be any of the remaining 2 digits. So, we have 2 choices for the fourth digit.
Step 2: Multiply the number of choices:
To find the total number of positive integers, we multiply the number of choices for each digit together: 5 × 4 × 3 × 2 = 120.
Step 3: Subtract the number of positive integers exceeding 4000:
Since we need to find positive integers less than 4000, we need to subtract the cases where the first digit is 5 or 6. In this case, the first digit can't be 5 or 6, so we have 3 choices for the first digit. The remaining digits can be selected as before: 4 × 3 × 2 = 24.
Step 4: Calculate the final result:
To obtain the number of positive integers less than 4000 with no repeated digits, we subtract the number of cases exceeding 4000 from the total number of cases: 120 - 24 = 96.
Therefore, the total number of positive integers less than 4000 with no repeated digits, using only the digits 2, 3, 4, 5, and 6, is 96.