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

How many ways can 5 different colored balls be placed in 3 different boxes if each box can hold any number of balls?

How many ways can 5 different colored balls be placed in 3 different boxes if each box can hold any number of balls?

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: Realize that each ball can go into any of the 3 boxes.
Step 4: For the first ball, there are 3 choices (Box 1, Box 2, or Box 3).
Step 5: For the second ball, there are also 3 choices (Box 1, Box 2, or Box 3).
Step 6: This pattern continues for all 5 balls, meaning each ball has 3 choices.
Step 7: To find the total number of ways to place all 5 balls, multiply the number of choices for each ball together: 3 choices for Ball 1, 3 choices for Ball 2, and so on.
Step 8: This can be expressed mathematically as 3 (choices for Ball 1) raised to the power of 5 (the number of balls), which is 3^5.
Step 9: Calculate 3^5, which equals 243.
Step 10: Conclude that there are 243 different ways to place the 5 different colored balls into the 3 boxes.

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 arrangements using powers, as each ball can independently go into any of the boxes.