Question
In how many ways can b blue balls and r red balls be distributed in n distinct boxes?
My Approach
It is clear that question is of the form of
Distributing indistinguishable objects into distinguishable boxes
given,
number of objects=$b+r$
number of boxes=$n$
So number of ways =$$\binom{(b+r)+n-1}{n-1}$$
But in the solution they are finding independently for red and blue balls and then multiplying i.e
Number of ways for red ball=$$\binom{r+n-1}{n-1}$$
Number of ways for blue ball=$$\binom{b+n-1}{n-1}$$
So, total number of ways=Number of ways for red ball=$$\binom{r+n-1}{n-1}*\binom{b+n-1}{n-1}$$
Which one is correct?
Best Answer
The given solution is correct. Your solution is incorrect because you do not have $b+r$ indistinguishable objects. Any of the $b$ blue socks is distinguishable from any of the $r$ red socks.
And the given solution is correct, since with no limit on the sizes of the boxes, you can think of this problem as first putting in the red socks, and then putting in the blue socks. That is, the number of ways I can distribute the blue socks is not at all affected by the number of ways I can distribute the red socks, and for every different way I distribute the red socks I can still distribute all the blue socks in all the different number of ways as if I had just blue socks. So, you multiply the number of ways you can distribute the red socks with the number of ways you can distribute the blue socks. And for each of those two tasks, you do use the 'putting indistinguishable objects into distinguishable boxes' formula.