The number of ways of giving 20 distinct oranges to 3 children such that each child gets at least one orange is \_\_\_\_\_\_\_\_\_\_\_.
Answer: 3483638676
Solutions · 1
Official solution
Total ways without any restrictions :
There are 320 ways to distribute the oranges to the 3 children.
Number of ways one child receives no orange :
Choose 1 child out of the 3 to not receive any orange in 3C1=3 ways. Distribute 20 oranges to the remaining 2 children in 220 ways. However, we've included the scenarios where the 2 children each get all the oranges. So, we subtract the 2 ways where one of the two remaining children gets all the oranges.
3C1(220−2)
Number of ways two children receive no orange :
Choose 2 children out of the 3 to not receive any oranges in 3C2=3 ways. The third child will receive all 20 oranges in 120=1 way.
3C2×120=3
Number of ways
= Total −( One child receive no orange + two child receive no orange)
=320−(3C1(220−2)+3C2120)=3483638676