Combinations and Permutations in ice cream cones... what is the difference? The first two I am certain I have correct however c and d... I am struggling to understand the difference. I have done research on this site and have seen similar questions with explanations but I just want to double check my thinking. 
A certain store sells 31 different flavors of ice cream.  How many different 3-scoop cones are possible if:
a. each flavor must be different and the order of the flavors is unimportant? $31 nCr 3 = 4495$
b. each flavor must be different and the order of the flavors is important? $31 nPr 3 = 26970$
c. Flavors need not be different and the order of the flavors is unimportant? (This is a nontrivial question.)
I think I need to use this formula $$\binom{n+m-1}{m}$$
where $n$ in the total numbers in the set, and m is how many numbers you want to choose.
So,  $$\binom{31+3-1}{3}=\binom{32}{3}=\frac{32\times 31\times 30}{1\times 2\times 3}=4960$$  Is this correct?
d. Flavors need not be different and the order of the flavors is important? 
I am thinking this is $31 \cdot 31 \cdot 31 = 29791$
 A: I think your (c) is wrong. You can count it by considering $3$ different cases: 1) $3$ flavors $X_3$; 2) $2$ flavors $X_2$ and 3) $1$ flavor $X_1$.


*

*The case of $3$ flavors is easy:
$$
X_3 = {31 \choose 3}
$$

*The case of $2$ flavors. You first choose $2$ flavors, then choose one of them to be the $2$ cones and another as the remaining cone.
$$
X_2 = {31 \choose 2}\cdot 2
$$

*The case of $1$ flavor.
$$
X_1 = {31 \choose 1}
$$
A: part d) is correct
part c) should be:
all different: $\binom{31}3$
two the same: $31\times30$
all the same 31
total $5456$
A: For c, you may use star-and-bar model, that is
$$\Large\star \star \star \normalsize \underbrace{||...||}_{30},$$
with $3$ stars and $30$ bars. The stars represent the $3$ scoops and the bars form $31$ slots, which represent the $31$ flavors. As the flavor can be the same and the order of the scoops is unimportant, there are 
$${31+3-1 \choose 3}={33 \choose 3}$$ ways, which is the same as the sum of the $3$ cases which Echo presents. In addition, there seems to be a minor math error in your answer for c.
A: (d) is correct.
For (c), you had the correct idea and formula, but you made an arithmetic error. $31 + 3 - 1$ is $33$, not $32$. Using $\binom{33}{3}$ would give you $5456$, which is the correct answer.
