In this article, we are going to learn how to calculate the number of ways in which x balls can be distributed in n boxes. This is one confusing topic which is hardly understood by students. But once mastered, it is the easiest topic of Permutation and Combination.

There can be 4 cases pertaining to this problem.

Case 1: Balls are same; boxes are same

Case 2: Balls are same; boxes are different

Case 3: Balls are different; boxes are same

Case 4: Balls are different; boxes are different

To understand it better, let's take an example.

Case 1: Balls are same; boxes are same

Case 2: Balls are same; boxes are different

Case 3: Balls are different; boxes are same

Case 4: Balls are different; boxes are different

To understand it better, let's take an example.

- Balls are same; boxes are same
- Balls are same; boxes are different
- Balls are different; boxes are same
- Balls are different; boxes are different

Group | Permutation of balls (Number of ways of grouping) | Ways of Distribution of boxes | Total number |

0,0,5 | 1 | 1 | 1*1=1 |

0,1,4 | 1 | 1 | 1*1=1 |

0,2,3 | 1 | 1 | 1*1=1 |

1,1,3 | 1 | 1 | 1*1=1 |

1,2,2 | 1 | 1 | 1*1=1 |

Hence, total number of ways = 1+1+1+1+1=5.

- When Balls are same; boxes are same

Group | Permutation of balls(Number of ways of grouping) | Ways of Distribution of boxes | Total number |

0,0,5 | 1 | 3!/2!=3 | 1*3=3 |

0,1,4 | 1 | 3!=6 | 1*6=6 |

0,2,3 | 1 | 3!=6 | 1*6=6 |

1,1,3 | 1 | 3!/2!=3 | 1*3=3 |

1,2,2 | 1 | 3!/2!=3 | 1*3=3 |

Hence, total number of ways = 3+6+6+3+3=21.

- Balls are same; boxes are different

Group | Permutation of balls(Number of ways of grouping) | Ways of Distribution of boxes | Total number |

0,0,5 | 1 | 1 | 1*1 |

0,1,4 | ^{5}C_{1}*^{4}C_{4}=5 | 1 | 5*1=5 |

0,2,3 | ^{5}C_{2}* ^{3}C_{3}=10 | 1 | 10*1=10 |

1,1,3 | ^{5}C_{3}* (2!/2)=10 | 1 | 10*1=10 |

1,2,2 | ^{5}C_{1} *(4!/(2!*2!*2)=15 | 1 | 15*1=15 |

Hence, total number of ways = 1+5+10+10+15=41

- Balls are different; boxes are same

Group | Permutation of balls(Number of ways of grouping) | Ways of Distribution of boxes | Total number |

0,0,5 | 1 | 3!/2!=3 | 1*3=3 |

0,1,4 | ^{5}C_{1} = 5 | 3!=6 | 5*6=30 |

0,2,3 | ^{5}C_{2} = 10 | 3!=6 | 10*6=60 |

1,1,3 | ^{5}C_{3}* (^{2}C_{1}/2) = 10 | 3!=6 | 10*6=60 |

1,2,2 | ^{5}C_{1} *(4!/2!*2!*2)=15 | 3!=6 | 15*6=90 |

Hence, total number of ways = 3+30+60+60+90= 243.

- Balls are different; boxes are different

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