Question

Let r and n be positive integers such that 1 ≤ r ≤ n. Then prove the following:

 ⁿC_r + 2 · ⁿC_{r − 1} + ⁿC_{r − 2} = ^n + 2C_r.

Answer 1

\[{}^n C_r + 2 . {}^n C_{r - 1} +^n C_{r - 2} = {}^{n + 2} C_r\]

\[LHS =^n C_r + 2 .^n C_{r - 1} +^n C_{r - 2} \]
\[ = \left( {}^n C_r +^n C_{r - 1} \right) + \left( {}^n C_{r - 1} +^n C_{r - 2} \right)\]

\[= {}^{n + 1} C_r + {}^{n + 1} C_{r - 1}\] [∵ \[n_{C_r} + n_{C_{r - 1}} = n + 1_{C_r}\]]

\[=^{n + 2} C_r\] [∵\[n_{C_r} + n_{C_{r - 1}} = n + 1_{C_r}\]]

 = RHS
∴  LHS = RHS