Permutation and Combination: Distribution of Balls into Boxes

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.
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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.
Example: What is the number of ways in which you can distribute 5 balls in 3 boxes when:
  • Balls are same; boxes are same
  • Balls are same; boxes are different
  • Balls are different; boxes are same
  • Balls are different; boxes are different
Solution:
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 5C1*4C4=5 1 5*1=5
0,2,3 5C2* 3C3=10 1 10*1=10
1,1,3 5C3* (2!/2)=10 1 10*1=10
1,2,2 5C1 *(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 5C1 = 5 3!=6 5*6=30
0,2,3 5C2 = 10 3!=6 10*6=60
1,1,3 5C3* (2C1/2) = 10 3!=6 10*6=60
1,2,2 5C1 *(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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