An art collection on auction consisted of 4 Dalis, 5 van Goghs, and 6 Picassos. At the auction were 5 art collectors. If a reporter noted only the number of Dalis, van Goghs, and Picassos acquired by each collector, how many different results could have been recorded if all of the works were sold?

The answer in my book is 8C4 x 9C4 x 10C4. But I don't understand why. Can anyone help to explain? Thanks.

Well there is a handy formula: The number of ways of choosing k things from n with replacement is: $\displaystyle n+k-1 \choose k$ See Wikipedia. Note that because it is with replacement there is nothing wrong with $\displaystyle n<k$. Either can be any non-negative number.

Now if you ask the question, how many ways are there to distribute n objects to k bins (people, etc.) you can think of the problem on its head. Each of n objects are going to pick a bin (people, etc.). Since 2 different objects could pick the same bin, this is picking n from k with replacement. So this is $\displaystyle n+k-1 \choose n$ or equivalently $\displaystyle n+k-1 \choose k-1$

So going back to your question: You are going to distribute 4 Dalis to 5 people: $\displaystyle 4+5-1 \choose 5-1$. You are going to distribute 5 VG's to 5 people: $\displaystyle 5+5-1 \choose 5-1$. And of course: $\displaystyle 6+5-1 \choose 5-1$

Since you can think of each collection by the same painter independently you just multiply them.