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 $31 \cdot 31 \cdot 31 = 29791$
Best Answer
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. $$ X_1 = {31 \choose 1} $$