MHB Can You Help Prove This Combinatorial Identity?

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Dear All,

I am trying to prove the following identity:

\[\binom{n}{k}=\binom{n-2}{k}+2\binom{n-2}{k-1}+\binom{n-2}{k-2}\]

My attempt was based on transforming the binomial coefficients into fractions with factorials, and then elimintating similar expressions. Somehow it didn't work out.

I believe that this proof shouldn't be too long. Can you assist ?Thank you in advance.
 
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Using Pascal's identity,
\[
\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}=\left[\binom{n-2}{k-2}+\binom{n-2}{k-1}\right]+\left[\binom{n-2}{k-1}+\binom{n-2}{k}\right]=\binom{n-2}{k-2}+2\binom{n-2}{k-1}+\binom{n-2}{k}.
\]
The identity itself has both algebraic and combinatorial proofs.
 
Thank you !
 
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