How many ways to distribute M different objects to N places where each places can receive from 0 to M objects

combinationscombinatoricspermutationsprobability

Here, assume $M > 1$

There are 3 cases

1. M < N
2. M = N
3. M > N

Each places can receive from 0 to M objects.

EDIT: example

M:1,2,3

N:5

Some possible combinations:

1,2,3,0,0

{1,2,3},0,0,0,0

{1,2},0,0,0,3

....

Here, each M components are different, so

$1,2,3,0,0 \neq 2,1,3,0,0$

But,

$\{1,2,3\},0,0,0,0 = \{2,1,3\},0,0,0,0=\{3,1,2\},0,0,0,0=…$

Therefore, once we consider more than 1 objects together, we do not consider the order of the objects in that particular group.

Repetition is not allowed.

Best Answer

Distributing M (different) objects to N (different) places/bins, repetition allowed, is equivalent to filling M (different) positions from N (different) choices, repetition allowed.

By the multiplication principle, there are $N^M$ ways this can happen.

A concrete example: there are $4^7$ ways for $7$ prizes to be distributed among $4$ participants, and there are $7^4$ ways for $4$ prizes to be distributed among $7$ participants.

P.S. On the other hand, distributing M identical objects to N (different) places/bins, repetition allowed, is a different story.

Related Question