In how many ways can $5$ red and $4$ white balls be chosen from a bag containing $10$ red and $8$ white balls?

Question

The correct answer is $\binom{10}{5}\binom{8}{4}$. However from the data given in the question , aren’t the balls identical? So the number of ways should actually be only $1$ right ?

Answer 1

It depends. If the balls are identical, then the answer is $1$, yes. But that's kindof boring, isn't it? I'm no mind reader, but it's obviously not the intended solution (unless it's a trick question). On the other hand, going directly from the choice of words and phrasing without putting any thought into it, I would likely have interpreted it your way.

I would personally have written "10 numbered red balls and 8 numbered white balls" or something like it if I had given this problem to students. As it stands, yes, the problem is ambiguous, and that's not good.