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After a round of "trick or treating", Candice has 13 KitKats and 14 Twix in her pillow case. Her mother asks her to share some (but not necessarily all) of the loot with her three younger brothers.

(A) How many different ways can she do this?

(B) How many different ways can she do this if she gives at least one of each type of bar to each of her brothers?

I'm unsure how you're meant to tackle this exactly. In my attempted solution (For A) I tried using the formula, n+k-1 C k-1, for the KitKats ( 15C3 ) and adding it to the Twixes (16C3) however my answer was incorrect.

Jacob Duggan
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    Two issues. $\binom{16}{3}$ is the number of ways to distribute $13$ kitkats among three brothers where she doesn't necessarily give all of her kitkats away, not $\binom{15}{3}$. Second, why would you add the results? Why wouldn't you multiply instead? – JMoravitz Oct 17 '17 at 19:05
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    You add in a situation where you break a scenario down into multiple non-overlapping cases, for example to count the number of students in a room you may add the number of boys to the number of girls to get the total. You multiply in a situation where when describing a scenario, you break it into steps, each step describing different aspects of it. For example, the number of outfits you can wear if you have five different shirts and three different pairs of pants (an outfit consisting of a specific shirt-pant pair). – JMoravitz Oct 17 '17 at 19:08
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    One more issue that I missed the chance to edit into my first comment, $\binom{16}{3}$ counts the number of ways of distributing the thirteen kitkats amongst herself and her brothers, but that includes her getting all of the kitkats. Similarly we might accidentally give her all of the twix as well. That wouldn't count as "sharing some" so we need to correct the final count once more. – JMoravitz Oct 17 '17 at 19:12

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