For 2 <= r <= n,binomnr + 2binomnr - 1 + binomnr - 2 =

(c) binomnr+2binomnr-1+binomnr-2=[binomnr+binomnr-1] +[binomnr-1+binomnr-2] \ =binomn+1r+binomn+1r-1 = binomn+2r [using binomnr + binomnr-1 = binomn+1r ]

For (2 leq r leq n,binom{n}{r} + 2binom{n}{r - 1} + bin... | Permutations and Combinations (Mathematics)

(c)

\binom{n}{r}+2\binom{n}{r-1}+\binom{n}{r-2}=\left[\binom{n}{r}+\binom{n}{r-1}\right] \\

+\left[\binom{n}{r-1}+\binom{n}{r-2}\right] \\ =\binom{n+1}{r}+\binom{n+1}{r-1} \\

\begin{aligned} &= \binom{n+2}{r} \quad \left[\text{using} \binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r} \right] \end{aligned}

Explanation

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