To determine the number of different orders in which 3 cards can be lined up, we need to use the concept of permutations. A permutation is an arrangement of all the members of a set into some sequence or order.
Here's how we approach this problem:
1. Identify the number of items to arrange: In this case, we have 3 cards.
2. Understanding permutations: The number of permutations of [tex]\( n \)[/tex] distinct items is given by [tex]\( n! \)[/tex] (n factorial), which means the product of all positive integers up to [tex]\( n \)[/tex].
3. Factorial Calculation: Factorial of a number [tex]\( n \)[/tex] (written as [tex]\( n! \)[/tex]) is defined as:
[tex]\[
n! = n \times (n-1) \times (n-2) \times \ldots \times 1
\][/tex]
Specifically, for 3 cards ([tex]\(n = 3\)[/tex]):
[tex]\[
3! = 3 \times 2 \times 1 = 6
\][/tex]
4. Result: Therefore, the number of different ways to line up 3 cards is [tex]\( 6 \)[/tex].
So, the correct answer is:
- [tex]\( 6 \)[/tex]
Hence, there are 6 different orders in which you can line up 3 cards on a table.
