determine the number of r-combinations of the multiset Determine the number of r-combinations of the multiset $\{1  \cdot a_1, \infty.  \cdot a_2, ... , \infty \cdot a_k \}$.
The answer in the back of the book is $\binom{r+k-2}{k-2}+\binom{r+k-3}{k-2}$.
I see where $\binom{r+k-2}{k-2}$ comes from because it is the number of r-combinations from the multiset that does not include $a_1$. I don't understand where the $\binom{r+k-3}{k-2}$ comes from though. 

As per the question, each element ranging from 'a2' to 'ak' can be included infinite number of time in 'r-combination' so produced, but element 'a1' can be included only once if required.
So now consider the following two cases :

Case 1 : If 'a1' is included in r-combination :

Then find number of ways of producing 'r-1'-combination from left 'k-1' elements which is equal to second term (r+k-3)C(k-2). 

Case 2 : If 'a1' is not included in r-combination :

Then find number of ways of producing 'r'-combination from left 'k-1' elements which is equal to first term (r+k-2)C(k-2). 
Summation of results produced in two cases produces final answer.
