How many ways are there to distribute 2 indistinguishable white and 4 indistinguishable black balls into 4 indistinguishable boxes?
How can we solve this?
[Math] How many ways are there to distribute 2 indistinguishable white and 4 indistinguishable black balls into 4 indistinguishable boxes
combinatoricsprobabilityprobability distributionsprobability theory
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Best Answer
There are 5 essentially different ways to distribute the black balls. In each case I'll count the essentially different ways of distributing the white balls.
The possible distinguishable ways of distributing the white balls are: 2-0-0-0, 1-1-0-0, 0-2-0-0 and 0-1-1-0. So 4 is the number.
Here we can do it like this: 2-0-0-0, 1-1-0-0, 1-0-1-0, 0-2-0-0, 0-1-1-0, 0-0-2-0, 0-0-1-1. So 7.
Once again: 2-0-0-0, 1-1-0-0, 1-0-1-0, 0-0-2-0, 0-0-1-1. 5 ways.
And again: 2-0-0-0, 1-1-0-0, 1-0-0-1, 0-2-0-0, 0-1-1-0, 0-1-0-1, 0-0-0-2. 7 ways.
Lastly: 2-0-0-0, 1-1-0-0. 2 ways.
In total, $4+7+5+7+2 = 25$ ways to distribute the balls.