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Ive searched for this problem but couldnt find it anywhere the way it is described like I have it, so Id be glad if you could help me with this.

Pat is to select three cookies from a tray containing 5 chocolate chip, 7 oatmeal and 4 peanut butter cookies. How many different assortments of three cookies can be selected?

Thanks for your help.

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    What do you define as an assortment? Is order of selection important at all? Are all of the chocolate chip cookies identical or do they have unique identification numbers? Does the question change at all if we replace the number $5$ with the phrase "*an unlimited supply of*"? (*the numbers $5,7,4$ are all extraneous information that doesn't matter in the slightest except to notice that regardless which three cookies pat takes there will be no problem, unlike in the situation where only two peanutbutter cookies were available and she couldn't have all three she chooses be pb*) – 2017-01-16
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    If order of selection matters, approach via multiplication principle or view this as the set of functions from $\{1,2,3\}$ to $\{chocchip,~oatm,~pb\}$. If order of selection doesn't matter, approach via [stars-and-bars](https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)). – 2017-01-16

2 Answers 2

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I'll assume that when you say "different assortments," you don't count oatmeal, oatmeal, peanut butter as different from oatmeal, peanut butter, oatmeal. That seems to be the most natural interpretation.

In that case, it's easy enough just to count the possibilities (with $P$ representing peanut butter, $O$ representing oatmeal, and $C$ representing chocolate chip):

$PPP$, $OOO$, $CCC$, $PCC$, $POO$, $OCC$, $OPP$, $COO$, $CPP$, and $COP$.

Answer: 10 Assortments

(Notice that the exact number of each cookie is irrelevant. As long as there are more than 2 of each type, we get these same assortments.)

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You could just count out the possible combinations by writing out a sequence of letters.