How many ways can 5 different colored balls be placed in 3 different boxes?

How many ways can 5 different colored balls be placed in 3 different boxes?

Correct Answer: 243

Step 1: Understand that we have 5 different colored balls.
Step 2: Recognize that there are 3 different boxes to place the balls in.
Step 3: For each ball, think about the choices available. Each ball can go into any of the 3 boxes.
Step 4: Since there are 5 balls and each ball has 3 choices, we can calculate the total number of ways by multiplying the choices for each ball.
Step 5: This means we have 3 choices for the first ball, 3 choices for the second ball, and so on, up to the fifth ball.
Step 6: The total number of ways to place the balls is calculated as 3 (choices for the first ball) multiplied by 3 (choices for the second ball) multiplied by 3 (choices for the third ball) multiplied by 3 (choices for the fourth ball) multiplied by 3 (choices for the fifth ball).
Step 7: This can be simplified using exponents: 3^5, which means 3 multiplied by itself 5 times.
Step 8: Calculate 3^5, which equals 243.
Step 9: Therefore, the total number of ways to place the 5 different colored balls in 3 different boxes is 243.

Counting Principles – This problem tests the understanding of the multiplication principle in combinatorics, where each choice is independent.
Exponential Growth – The solution involves calculating the total number of combinations using powers, as each ball has multiple independent choices.