MHB Can You Help Prove This Combinatorial Identity?

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The discussion focuses on proving the combinatorial identity \(\binom{n}{k}=\binom{n-2}{k}+2\binom{n-2}{k-1}+\binom{n-2}{k-2}\). The initial approach involved transforming binomial coefficients into factorials, but it was unsuccessful. A suggestion was made to utilize Pascal's identity to derive the proof, demonstrating the relationship between the coefficients. The identity can be validated through both algebraic and combinatorial methods. Assistance is requested to finalize the proof effectively.
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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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