Choose Sixteen Cookies from Five Varieties A cookie shop sells 5 different kinds of cookies. How many different ways are there to choose 16 cookies if...
(a) you have no restrictions?
(b) you pick at least two of each? 
(c) you pick at least 4 oatmeal cookies, at least 3 sugar cookies and at most 5 chocolate chip cookies?
This is what I have been trying: For part a) I simply count 5 ways for the first slot, 5 ways for the second... continued to $$ 5^{16} $$
part b... This is throwing me off. I do not understand how to implement the condition. would I reserve two spots such as 
$ {4^2} + 5 ^{14} $?
part c... Is it the same principle as above? 
 A: A weak composition of $n$ into exactly $k$ parts is an ordered $k$-tuple of non-negative integers whose entries sum to $n$. For example, $(5, 4, 2, 0, 5)$ is a weak composition of $16$ into exactly five parts. 
We can associate this weak composition with a selection of cookies: we purchased 5 cookes of type A, 4 cookies of type B, 2 cookies of type C, 0 cookies of type D, and 5 cookies of type E. In fact, the number of compositions of 16 into exactly five parts counts precisely the number of ways to purchase the cookies. Using the method of Stars and Bars, one could prove the number of weak compositions of $n$ into exactly $k$ parts is $\binom{n+k-1}{k-1}$, which is $\binom{20}{4} = 4,845$ in this instance.
For part (b), begin by taking two of each cookie. You have no freedom here. Now, it remains to select 6 more cookies with no restrictions. How many ways can this be accomplished?
Handle part (c) similarly to part (b), but be careful to subtract away the selections that have too many chocolate chip cookies.
