To solve problem number 6, let's break down the question and calculate each part systematically.

The question provides a set of digits $\{1, 2, 3, 4, 5, 6, 7, 8\}$ to form four-digit numbers, each using different digits.

The tasks are:

1. Count the number of four-digit numbers less than 3000.
2. Count the number of four-digit numbers between 3000 and 6000.
3. Count the number of four-digit numbers greater than 6000.

Let's solve each part.

Part a: Four-digit numbers less than 3000

For a four-digit number to be less than 3000, the thousands place must be either 1 or 2.

1. If the thousands digit is 1:

   • We have 7 choices remaining for the hundreds place (2, 3, 4, 5, 6, 7, 8).
   • After choosing the hundreds place, 6 choices remain for the tens place.
   • Finally, we have 5 choices for the units place.
   • Total numbers with 1 in the thousands place: $7 \times 6 \times 5 = 210$.

2. If the thousands digit is 2:

   • Similar calculation: $7 \times 6 \times 5 = 210$.

Thus, the total number of four-digit numbers less than 3000 is: $210 + 210 = 420$

Part b: Four-digit numbers between 3000 and 6000

For a number to be between 3000 and 6000, the thousands place must be either 3, 4, or 5.

1. If the thousands digit is 3:

   • Choices for the hundreds, tens, and units places: $7 \times 6 \times 5 = 210$.

2. If the thousands digit is 4:

   • Choices for the hundreds, tens, and units places: $7 \times 6 \times 5 = 210$.

3. If the thousands digit is 5:

   • Choices for the hundreds, tens, and units places: $7 \times 6 \times 5 = 210$.

Thus, the total number of four-digit numbers between 3000 and 6000 is: $210 + 210 + 210 = 630$

Part c: Four-digit numbers greater than 6000

For a number to be greater than 6000, the thousands place must be either 6, 7, or 8.

1. If the thousands digit is 6:

   • Choices for the hundreds, tens, and units places: $7 \times 6 \times 5 = 210$.

2. If the thousands digit is 7:

   • Choices for the hundreds, tens, and units places: $7 \times 6 \times 5 = 210$.

3. If the thousands digit is 8:

   • Choices for the hundreds, tens, and units places: $7 \times 6 \times 5 = 210$.

Thus, the total number of four-digit numbers greater than 6000 is: $210 + 210 + 210 = 630$

Final Answer

• a. Four-digit numbers less than 3000: 420
• b. Four-digit numbers between 3000 and 6000: 630
• c. Four-digit numbers greater than 6000: 630

Would you like more details on the solution steps or have any questions?

Here are some additional related questions:

1. How would the solution change if repetition of digits were allowed?
2. What if we are restricted to using only even numbers in each place?
3. How many numbers could be formed if the number had to be a multiple of 5?
4. Can we use the same approach to form three-digit numbers with different conditions?
5. How would the calculations adjust if zero was included as an option for digits?

Tip: When forming numbers with a specific range, focus first on the highest place value (like thousands in this case) to narrow down your choices effectively.