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How many ways are there to choose 16 cookies if there are six varieties of cookies including chocolate chip, and at least six chocolate chip cookies must be chosen?

Is $C(n+r-1,n-1)$ correct where $n=16$ and $r=6$?

(There is an unlimited amount of each typy of cookie)

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    $\begingroup$ Are you assuming unlimited quantity of each type of cookie? $\endgroup$ – jdods Aug 5 '15 at 15:08
  • $\begingroup$ @jdods Yes there is $\endgroup$ – Julia Aug 5 '15 at 15:10
  • $\begingroup$ Do you mean permutations (specific orderings) or combinations ? $\endgroup$ – David Quinn Aug 5 '15 at 15:10
  • $\begingroup$ @DavidQuinn Combinations $\endgroup$ – Julia Aug 5 '15 at 15:10
  • $\begingroup$ So now the question reduces to choosing $10$ cookies out of $6$ varieties, with no restrictions. Can you handle this ? $\endgroup$ – Shailesh Aug 5 '15 at 15:11
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$Hint:$

First take out the $6$ chocolate chip cookies anyways. Now the problem reduces to choosing $10$ cookies from $6$ varieties with no restriction, which is the stars and bars problem, with $n=10$ and $r = 6$

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  • $\begingroup$ So is $C(n+r-1,n-1)$ correct with $n=10$ and $r=6$. I made a mistake, I meant to ask it this way, but I do not want to change the question now. $\endgroup$ – Julia Aug 5 '15 at 15:22
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    $\begingroup$ The second parameter should be $r - 1$. Otherwise it sees OK. That was probably just an oversight. $\endgroup$ – Shailesh Aug 5 '15 at 15:23
  • $\begingroup$ I like to turn it around and think of it as throwing 10 balls into 6 buckets, and considering how many ways there is to do that. I find it easier to think of it this way. The 6 buckets represent the types of cookies, and we are choosing a total of 10. Personally, I can more easily make sense of this as a multiset/stars-&-bars problem. $\endgroup$ – jdods Aug 5 '15 at 15:50

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