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Prove combinatorially that (10 choose 4) = (9 choose 3) + (9 choose 4)

 Mar 15, 2021
 #1
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10 c 4 =   10! / (4! 6!) = 210

 

9 c3 = 9! / (3! 6!) = 84

9 c 4 = 9!/ ( 4! 5!) = 126       126 + 84 = 210

 Mar 15, 2021
 #2
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Is there a way to do this without calculating it? Through combinitoric means?

Guest Mar 15, 2021
 #3
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we have the hockeystick identity which gives you instantly the answer

 

we can prove the hockeystick identity by using trianglular numbers

jxc516  Mar 16, 2021

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