The first thing to realize is that the two problems are the same: each way of filling the doughnut order corresponds to a unique solution of $x_1+\ldots+k_k=r$ in non-negative integers, and each such solution corresponds to a unique way of filling the doughnut order. Specifically, if the doughnuts are of types $T_1,\dots,T_k$, and for $i=1,\dots,k$ we let $x_i$ be the number of doughnuts of type $T_i$ that you get, then the numbers $x_i$ are certainly non-negative integers, and they satisfy the equation $x_1+\ldots+x_k=r$. Conversely, if $x_1,\dots,x_k$ are non-negative integers whose sum is $r$, we can make up the doughnut order with $x_1$ doughnuts of type $T_1$, $x_2$ of type $T_2$, and so on.
Imagine that you lay out the doughnuts in an order in a line, first all those of type $T_1$, then all those of type $T_2$, and so on. To make sure that you keep the types straight, you put atoothpicks between the types. For instance, if $k=4$ and $r=10$, you might have $3$ doughnuts of type $T_1$, $2$ of type $T_2$, and $5$ of type $T_3$, and your string of doughnuts would look like this: OOO|OO|OOOOO
. You might have no doughnuts of type $T_1$, $4$ of type $T_2$, and $6$ of type $T_3$: |OOOO|OOOOOO
. Or you might have $2$ doughnuts of type $T_1$, none of type $T_2$, and $8$ of type $T_3$: OO||OOOOOOOO
. Every such string will contain $10$ doughnuts and $2$ toothpicks. More generally, if you have $k$ types of doughnuts and $r$ doughnuts altogether, each string will contain $r$ doughnuts and $k-1$ toothpicks, because there are only $k-1$ division points between adjacent types.
Every doughnut order corresponds to one of these strings, and every string of $r$ doughnuts and $k-1$ toothpicks corresponds to a possible doughnut order, so to count the possible doughnut orders (and hence also the solutions in non-negative integers to the equation $x_1+\ldots+x_k=r$) we need only count the possible strings of $r$ doughnuts and $k-1$ toothpicks. Each is a string of $r+k-1$ objects. Once we know where the $r$ doughnuts go in the string, the $k-1$ toothpicks must occupy the other $k-1$ positions. There are $\binom{r+k-1}r$ ways to choose $r$ of the $r+k-1$ positions to be occupied by the $r$ doughnuts, so the number of possible strings is $\binom{r+k-1}r$. Alternatively, there are $\binom{r+k-1}{k-1}$ ways to choose $k-1$ positions for the toothpicks, after which the doughnuts will automaticall occupy the other $r$ positions, so there are $\binom{r+k-1}{k-1}$ possible strings. (Of course we know that $$\binom{r+k-1}r=\binom{r+k-1}{k-1}\;,$$ so there’s no conflict here.)
We’ve already seen that there are exactly as many strings of doughnuts and toothpicks as there are possible doughnut orders, and exactly as many doughnut orders as there are solutions in non-negative integers to the equation $x_1+\ldots+x_k=r$, so the answer to both problems is $$\binom{r+k-1}r=\binom{r+k-1}{k-1}\;.$$
Best Answer
Forget my smart ass comments. The book is assuming the boss has a list of employees (himself included so there are six employees) one after another. He buys donuts one by one each one specifically for a specific employee.
There are $12$ Types of donuts.
Each employee has a choose of $12$ types of donuts. There are $12^6$ ways to do this. (Don't forget he is buying a donut for himself as well)
$P(12, 6)$ there are $12$ chooses for the first employeed, $11$ for the second....etc.
"For this last question I'm having difficulty understanding the wording."
I don't see why. This simply a matter of choosing $6$ types of donuts from $12$ different types. THat is ${12 \choose 6} =\frac {12!}{6!6!}$
" Is the manager splitting the 12 donuts into 6 categories, therefore there are 2 types of donuts in each category"
Uh... no. He is doing nothing of the sort. He is buying $6$ (not $12$) donuts and putting them on a table and walking away.
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On rereading question 3 is badly worded "he wishes to select one donut of each of six different types" makes it sound as though the store only had $6$ types not $12$. But still I think it only makes sense as how bought six different types of donuts (one each).
And my smart ass comment is that in buying the donuts order doesn't matter. It's only in passing out the donuts that order matters. But that's me being a smart ass.