Shaking Hands Problem
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In the casework method, \(10+9+8+...+1\) looks very familiar... it's the famous shaking hands problem!
Suppose you have 11 people shaking hands with each other, then it's \(10+9+...+1\) I'm pretty sure no explanation is needed.
How about the stars
and bars 
method? It turns out you can explain it using the more "fancy" trick too!
The number of ways to choose 2 out of 11 people to shake hands with each other is, well, \(\binom{11}{2}\).
Same as the stars and bars grade curve problem!
And if there are 4 grades, like in the mini-question...
You will have 3 people shaking hands with each other at the same time!
Weird, but true.
