Is the rule to always have something choose 2?
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[Originally posted in Discussions]
When Prof Po said 6 choose 2, where did he get that from?
so is the rule always x choose 2 or is there something behind it?
The term "6 choose 2" means "the number of ways to choose two things out of six different things." This came up in the lesson when Prof Loh was finding the answer to the question, "How many intersection points are there?"
To find out the value of 6 choose 2, it might help to color-code the six lines like this:
The 6 × 5 comes from the fact that there are 6 groups of ways, with 5 ways in each group. I'll illustrate two of the groups right here:
Group 1 (Ways start with Red)
Group 2: (Ways start with Green)
Group 3: (Ways start with Yellow) (not pictured)
Group 4: (Ways start with Blue) (not pictured)
Group 5: (Ways start with Black) (not pictured)
Group 6: (Ways start with Dark Green) (not pictured)
This is an easy way to count the ways, because without drawing the ways out, we can see that the 6 comes from the number of lines, and the 5 came from the number of other lines left over after we chose the first one! However, the downside to this easy way is that we have to consider the double-counted ways. In the Red Group and Green Group, there is a way that occurs in both: the intersection of the red line and the green line.
Actually, every way has a copy, and if you drew out all the lines for all the groups, you would see this!
So we divide 2 to the number of ways that we drew out, which gives
as the number of different intersection points. (Note that this was just a warm-up question; it was the answer to the hint. This is to show you the idea of how to choose x things from y things, and isn't needed for finding the number of triangles.)
I hope this helps! Let me know in the Discussions.
The Daily Challenge Team