In how many ways can b blue balls and r red balls be distributed in n distinct boxes.?

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?

• The second one is correct since blue balls are distinguishable from red balls. – N. F. Taussig Jun 28 '17 at 11:35

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.