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Question
Let r and n be positive integers such that 1 ≤ r ≤ n. Then prove the following:
nCr + 2 · nCr − 1 + nCr − 2 = n + 2Cr.
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Solution
\[LHS =^n C_r + 2 .^n C_{r - 1} +^n C_{r - 2} \]
\[ = \left( {}^n C_r +^n C_{r - 1} \right) + \left( {}^n C_{r - 1} +^n C_{r - 2} \right)\]
∴ LHS = RHS
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