Question

The number of ways of giving 20 distinct oranges to 3 children such that each child gets at least one orange is______ 

Answer 1

Correct Answer: 348

Total Cases:

\[ 3^{20} \] 

Subtracting Invalid Cases:

• Case 1: One child receives no orange: \[ \binom{3}{1} \cdot 2^{20 - 2} \]
• Case 2: Two children receive no orange: \[ \binom{3}{2} \cdot 1^{20} \]

Final Calculation:

\[ \text{Total} - (\text{One child receives no orange} + \text{Two children receive no orange}) \]

\[ = 3^{20} - \binom{3}{1} \cdot 2^{20 - 2} + \binom{3}{2} \cdot 1^{20} \]

\[ = 3483638676 \]