deletions | additions       diff --git a/As_you_can_see_we__.tex b/As_you_can_see_we__.tex  index 670bb80..a9d345e 100644  --- a/As_you_can_see_we__.tex  +++ b/As_you_can_see_we__.tex 
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As you can see, we had to create 4 additional placeholders for the bars. The bars divide the batarangs into who will be getting how many batarangs thrown at them. As you can see, after getting hit by their first batarangs, Villain 1 will get hit with 5 batarangs, Villain 2 will get hit by 3 batarangs, Villain 3 will get hit by 0 batarangs, Villain 4 will get hit by 6 batarangs and Villain 5 will get hit with 1 batarang in this particular arrangements. How many possible arrangements can there be just moving the bars around? Well, as you can see, we must take the total amount of placeholders and choose 4 spots for each of the bars. So by definition, that would be ${19}\choose{4}$, which works out to be $\frac{19!}{4!15!}$ different ways to get rid of all the batarangs. If we didn't have to hit each villain at least one time, we would just use the method of stars and bars from the very beginning of the problem so that would be ${24}\choose{4}$ ways because we would have 20 batarangs and 4 bars to divide them, so that makes 24 placeholders, and we must choose how to place each of the 4 bars.  Exam question:  The Incredible Hulk loves smashing things. He has a good 11 smashes left before changing back into Bruce Banner. In front of him is an old pickup truck, a soda vending machine, and a tractor. How many different ways are there for the Incredible Hulk to use his 11 smashes on these 3 objects? He doesn't have to hit all the objects.