To determine how many different ways you can line up 8 cards, we need to calculate the number of permutations of the 8 cards. A permutation is an arrangement of objects in a specific order.
The formula for permutations of n distinct objects is given by:
P(n) = n!
Where n! (n factorial) is the product of all positive integers up to n. For our 8 cards, we need to calculate 8!:
8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40320
This means there are 40,320 different ways to line up the 8 cards on a table. Each unique order represents a different permutation of the cards, demonstrating the numerous arrangements available even with just a small number of items.