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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.