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The proof I tried is in the aforementioned images, but I don't know how to proceed after a point. I would like a completed version of that proof or if it isn't viable, I would also love exploring other proofs along with a reason why my proof isn't a viable one.

an4s
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Santhosh Annamalai
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    Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures. – José Carlos Santos Jan 08 '20 at 10:26
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    please write it out instead of having pictures – Lukas Rollier Jan 08 '20 at 10:27
  • It is a rather well known result. Certainly already addressed in this site. Are you obliged to prove it by induction? – Damien Jan 08 '20 at 10:30
  • Hi! I'm a mobile user and would have used MathJax if I posted this question using a PC, because it is very inconvenient to type using mobile, sorry if I broke a rule – Santhosh Annamalai Jan 08 '20 at 10:32
  • No, I'm not obliged to prove it by induction, however the first method I tried for this problem was the induction method, but I didn't know how to complete it. I just thought of this induction proof as a learning opportunity – Santhosh Annamalai Jan 08 '20 at 10:34
  • A simpler proof than induction is provided here: https://math.stackexchange.com/questions/148583/combinatorial-proof-of-summation-of-sum-limits-k-0n-n-choose-k2-2n – Damien Jan 08 '20 at 10:36
  • Yeah I do know that proof, but I was just wondering if I could prove it by induction – Santhosh Annamalai Jan 08 '20 at 10:44
  • Thanks, and sorry for asking a duplicate question. I wasn't aware of that notation for combinations before so I couldn't search it here like that – Santhosh Annamalai Jan 08 '20 at 10:55

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