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MCII-B Combinatorics 8.24, 8.28, and 8.29

 
 
ZhangDaniel的头像
MCII-B Combinatorics 8.24, 8.28, and 8.29
ZhangDaniel - 2024年08月6日 Tuesday 21:16
 

I tried drawing a diagram for question 8.24, but I'm struggling on figuring out how to find the shaded regions. 

For 8.28, I drew a smaller hexagon within the big hexagon with side length 1, as I thought that would be the area that wouldn't work. I don't think this is right though...

For 8.29, I'm just kind of confused as to how I would solve it. Where could I start?

 
LensmireJohn的头像
Re: MCII-B Combinatorics 8.24, 8.28, and 8.29
LensmireJohn - 2024年08月7日 Wednesday 09:29
 

Here's some hiints/comments:

For 8.24, drawing a diagram is necessary, just be careful with the requirements. As a note to clarify, Jack will not wait after 5:45 means he definitely will not wait after 5:45 (even if he he started waiting at 5:43, he'll stop at 5:45). The region might be a little confusing, but it should be all straight lines, so areas can be calculated with triangles, parallelograms, etc.

For 8.28, let's consider a simpler question, what are all the points that are a distance of less than 1 from (0,0). Well the points of exactly distance one form a circle of radius 1, so points of distance less than 1 are all the points inside this circle. This type of reasoning will be needed for 8.28 because we're talking about distances from vertices.

For 8.29 we're picking 4 balls, so there should be (18 choose 4) total outcomes. The "Be careful here" hint is meant to make clear we need to consider some different cases to get at least one of each color.