How many ways can 4 different colored balls be arranged in a line?

How many ways can 4 different colored balls be arranged in a line?

Correct Answer: 24

Step 1: Understand that we have 4 different colored balls. Let's call them Ball A, Ball B, Ball C, and Ball D.
Step 2: We want to find out how many different ways we can arrange these 4 balls in a line.
Step 3: The first position in the line can be filled by any of the 4 balls. So, there are 4 choices for the first position.
Step 4: After placing one ball in the first position, we have 3 balls left. So, there are 3 choices for the second position.
Step 5: After placing balls in the first two positions, we have 2 balls left. So, there are 2 choices for the third position.
Step 6: Finally, there is only 1 ball left for the last position. So, there is 1 choice for the fourth position.
Step 7: To find the total number of arrangements, we multiply the number of choices for each position: 4 (for the first) × 3 (for the second) × 2 (for the third) × 1 (for the fourth).
Step 8: This multiplication gives us 4 × 3 × 2 × 1 = 24.
Step 9: The result, 24, is the total number of ways to arrange the 4 different colored balls in a line.

Factorial – The concept of arranging distinct objects is calculated using factorial notation, where n! represents the product of all positive integers up to n.
Permutations – The arrangement of different items in a specific order is known as permutations, which is relevant when considering the order of the colored balls.