How many ways can 4 different colored balls be placed in 3 different boxes?

How many ways can 4 different colored balls be placed in 3 different boxes?

Correct Answer: 81

Step 1: Identify the number of balls. We have 4 different colored balls.
Step 2: Identify the number of boxes. We have 3 different boxes.
Step 3: Understand that each ball can go into any of the 3 boxes. This means for each ball, there are 3 choices.
Step 4: Since there are 4 balls and each has 3 choices, we multiply the choices together. This is done by raising the number of choices (3) to the power of the number of balls (4).
Step 5: Calculate 3 raised to the power of 4, which is 3^4.
Step 6: Perform the calculation: 3^4 = 3 * 3 * 3 * 3 = 81.
Step 7: Conclude that there are 81 different ways to place the 4 different colored balls into the 3 different boxes.

Counting Principle – The problem utilizes the fundamental counting principle, where each choice (placing a ball) is independent of the others.
Combinatorics – The question involves combinatorial reasoning, specifically the arrangement of distinct items into distinct groups.