To determine the number of different orders in which you can line up 6 cards on a table, we need to calculate the factorial of 6. The factorial of a number [tex]\( n \)[/tex] (denoted as [tex]\( n! \)[/tex]) is the product of all positive integers up to [tex]\( n \)[/tex].
Here's the step-by-step process to find [tex]\( 6! \)[/tex]:
1. Start with the number 6.
2. Multiply 6 by each descending integer until you reach 1:
[tex]\[
6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1
\][/tex]
3. Perform the multiplications step-by-step:
- [tex]\( 6 \times 5 = 30 \)[/tex]
- [tex]\( 30 \times 4 = 120 \)[/tex]
- [tex]\( 120 \times 3 = 360 \)[/tex]
- [tex]\( 360 \times 2 = 720 \)[/tex]
- [tex]\( 720 \times 1 = 720 \)[/tex]
So, the number of different orders to line up 6 cards on a table is [tex]\( 720 \)[/tex].
Therefore, the correct answer is [tex]\( 720 \)[/tex].
