Why is the total number of ways to place 4 distinct balls into 4 distinct boxes $4^4$?

Conditions

• The balls are distinct.
• The boxes are distinct.
• Each ball must be placed into exactly one box.
• Empty boxes are allowed.

Reasoning, step by step

1. Consider the first ball: it has 4 possible boxes.
2. Consider the second ball: it also has 4 possible boxes, independent of the first.
3. Repeat for the third and fourth balls.
4. Multiply the number of choices for each ball: $4 * 4 * 4 * 4 = 4^4$.

Example

The video states: "Each of four distinct balls has four box choices, giving 4 to the power 4 individual assignments."

Common misconceptions

• Using 4! instead of $4^4$, which would imply no empty boxes.
• Using $C(4,4)$ or other combination formulas that ignore the distinctness of balls and boxes.

Watch the explanation

Connected concepts

Explore next

• How does this total count serve as the denominator in probability calculations?
• What is the probability of a specific single assignment?

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