Let's break this problem down step by step.

Part 1: Total number of four-digit numbers

We need to form four-digit numbers using the digits 4, 5, 6, and 9, with no repetition of any digits.

• Step 1: The first digit must be chosen from the available digits 4, 5, 6, or 9. So, there are 4 choices for the first digit.
• Step 2: After choosing the first digit, there are 3 remaining choices for the second digit.
• Step 3: For the third digit, after two digits are chosen, there are 2 remaining choices.
• Step 4: For the fourth digit, after three digits are chosen, there is 1 remaining choice.

Thus, the total number of four-digit numbers that can be formed without repetition is:

$4 \times 3 \times 2 \times 1 = 24$

Part 2: Four-digit numbers between 4000 and 6000

To ensure the number is between 4000 and 6000, the first digit must be 4 or 5. Let's break it down:

• Step 1: If the first digit is 4, then the remaining digits (5, 6, and 9) can be arranged in any order for the other three positions. There are 3 choices for the second digit, 2 choices for the third, and 1 choice for the fourth. So, there are:

  $3 \times 2 \times 1 = 6$

• Step 2: If the first digit is 5, then the remaining digits (4, 6, and 9) can also be arranged in any order for the other three positions. Again, there are:

  $3 \times 2 \times 1 = 6$

Thus, the total number of numbers between 4000 and 6000 is:

$6 + 6 = 12$

Final Answer:

• The total number of four-digit numbers that can be formed using the digits 4, 5, 6, 9 without repetition is 24.
• The total number of such numbers between 4000 and 6000 is 12.

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Do you want any additional explanations or further details on these calculations?

Here are five related questions you might find interesting:

1. How many four-digit numbers can be formed using any 4 digits from 0 to 9, without repetition?
2. What is the probability that a randomly chosen four-digit number from the above set is between 4000 and 6000?
3. If the digits could repeat, how many four-digit numbers can be formed?
4. What if the first digit had to be a number between 1 and 5, how many numbers could be formed?
5. How would the number of valid four-digit numbers change if the digits 4, 5, 6, and 9 were used but there was no restriction on the first digit?

Tip: When you're dealing with such problems, always consider constraints on the first digit first, as it often narrows down your choices significantly.