How many ways can 8 different colored balls be arranged in a line?

How many ways can 8 different colored balls be arranged in a line?

Step 1: Understand that we have 8 different colored balls.
Step 2: Realize that we want to arrange these 8 balls in a line.
Step 3: Know that the number of ways to arrange 'n' different items is given by 'n!'.
Step 4: Since we have 8 balls, we need to calculate 8!. This means we multiply all whole numbers from 1 to 8 together.
Step 5: Calculate 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1.
Step 6: Perform the multiplication step-by-step: 8 × 7 = 56, then 56 × 6 = 336, then 336 × 5 = 1680, then 1680 × 4 = 6720, then 6720 × 3 = 20160, then 20160 × 2 = 40320, and finally 40320 × 1 = 40320.
Step 7: Conclude that there are 40320 different ways to arrange the 8 different colored balls in a line.

Permutations – The question tests the understanding of permutations, specifically how to calculate the number of ways to arrange a set of distinct objects.