How is the counting formula $C_4^2(C_4^3C_2^1+C_4^2)$ for exactly two empty boxes derived?

Conditions

• The balls are distinct.
• The boxes are distinct.
• Exactly two boxes must be empty, meaning the other two must be non-empty.

Reasoning, step by step

1. Select 2 empty boxes from 4: $C_4^2$.
2. Identify the two possible occupancy patterns for the remaining 2 boxes: $3+1$ and $2+2$.
3. Calculate ways for $3+1$ pattern: Choose 3 balls from $4 (C_4^3)$ and choose which of the 2 occupied boxes gets them ($C_2^1$). The remaining 1 ball goes to the other box.
4. Calculate ways for $2+2$ pattern: Choose 2 balls from 4 for one specific box ($C_4^2$). The remaining 2 balls go to the other box.
5. Sum the patterns and multiply by the box selection: $C_4^2 * (C_4^3*C_2^1 + C_4^2)$.

Example

The video writes $P(B) = [C_4^2(C_4^3 C_2^1 + C_4^2)] / 4^4$.

Common misconceptions

• Forgetting to multiply by $C_2^1$ in the $3+1$ case, which assigns the group of 3 to a specific box.
• Double-counting in the $2+2$ case by treating the boxes as unlabeled (they are distinct, so choosing 2 balls for Box X is different from choosing them for Box Y).

Watch the explanation

Connected concepts

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• What is the numerical value of this expression?
• How does this numerator combine with the denominator $4^4$ to find $P(B)$?