To determine how many different 5-digit numbers can be formed using the digits 1, 2, 3, 4, 5, and 6, where each digit is unique, we need to calculate the number of permutations of 5 digits out of 6.

Step-by-Step Solution:

1. Choose the first digit: There are 6 options (1, 2, 3, 4, 5, 6).
2. Choose the second digit: Since the digits must be different, there are 5 remaining options.
3. Choose the third digit: There are 4 options left.
4. Choose the fourth digit: There are 3 options left.
5. Choose the fifth digit: There are 2 options left.

Calculation:

$6 \times 5 \times 4 \times 3 \times 2 = 720$

So, 720 different 5-digit numbers can be formed using the digits 1, 2, 3, 4, 5, and 6, where each digit is unique.

Would you like to explore more details or have any questions?

Here are 5 related questions:

1. How many 6-digit numbers can be formed with the digits 1, 2, 3, 4, 5, 6?
2. What if repetition of digits was allowed? How many 5-digit numbers could be formed?
3. How many 4-digit numbers can be formed with the digits 1, 2, 3, 4, 5, 6, without repetition?
4. How many odd 5-digit numbers can be formed with the digits 1, 2, 3, 4, 5, 6, without repetition?
5. How many 5-digit numbers can be formed using only the digits 1, 2, 3, 4, 5?

Tip: When calculating permutations, remember that order matters. For combinations, order does not matter.