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If there are 16 different ice-cream flavours, how many combinations are there for a two scoop?
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If there are still 16 different ice-cream flavours, how many combinations are there for a three scoop?
How would you work out the above combinations? I found it just sitting in my notes app and I don't recall I ever found an answer.
My thoughts at the moment are 16 * (no. of scoops) but I am still lost since of course the flavours can appear in any order. (As my SE profile will demonstrate, I'm not much of a mathematician!)
So what would an equation be for the following as applicable to both Problem 1 and Problem 2?
A. Working out the number of combinations including duplicate scoops (e.g. chocolate-chocolate-vanilla)
B. Working out the number of combinations where a flavour only appears once in each possible combination (e.g. chocolate-vanilla-strawberry and then vanilla-chocolate-strawberry).
Any help would be much appreciated!
Best Answer
1: This problem is equivalent to putting two identical balls into 16 boxes (with repetition allowed).
$\binom{2+16-1}{2}$
Where 2 is the number of balls, and $16-1$ is the number of separators needed to divided a line segment into 16 smaller segments,each segment represents a box.
Each way of putting the balls (each combination of flavors) will be corresponds to a arrangement of $2$ balls and $16-1$ seperators (where balls and seperators are indistinguishable among themselves).
2.$\binom{3+16-1}{3}$.
A. As shown above, where we assume repetition is allowed.
B. If repetition is not allow,
$\binom{16}{2}$
$\binom{16}{3}$