[Math] Number of ways can 3 objects be distributed in 5 boxes

combinatoricspermutationsproof-verification

I was solving a permutation question but my answer came out different from answer given in the book.
We can put more than 1 ball in a box.
My reasoning is:

  1. There are 5 ways of choosing 1 box. So there are 5 ways of distributing all 3 objects in 1 box.
  2. There are $5\cdot 4 = 20$ ways of choosing 2 boxes. And for every 1 way there are 3 ways of distributing objects in them. So there are total $20\cdot 3 = 60$ ways of distributing objects in 2 boxes.
  3. There are $5\cdot 4\cdot 3 = 60$ ways of choosing 3 boxes and 6 ways of distributing objects. So 360 ways.

So all over there are $5 + 60 +360 = 425$ ways of distributing 3 objects in 5 boxes.

Reasoning given in the book is:

There are 5 choices for every objects and so there are $5\cdot 5\cdot 5 = 125$ ways of distributing 3 objects in 5 boxes.

What is the problem with my reasoning.

Best Answer

The only mistake is in your third step.

There are $60$ ways to choose an ordered set of three boxes among the five. This fixes the order, so you do not need to multiply by $6$ again to place the balls. If instead you said that there are $\frac{5 \cdot 4 \cdot 3 }{ 6}$ ways to choose an unordered set of three boxes from the five, then you would need to multiply by $6$ to account for the ways you can put the three balls in the boxes.

Note that this problem did not arise in your second step. When you say there are $5 \cdot 4 = 20$ ways to choose two boxes, this is an ordered set of two boxes, so you can view it as implicitly saying "the first box will be the box with two balls, and the second one will be the box with one ball." Then you just multiply by $3$ to decide how to place the balls into these two boxes.

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