For 2 ≤ r ≤ n , ( n r ) + 2 ( n r − 1 ) + ( n r − 2 ) is equal to Options: A . &nbsp(n+1r−1) B . &nbsp(n+1r+1) C . &nbsp2(n+2r) D . &nbsp(n+2r)

Answer: Option D : D ( n r ) + 2 ( n r − 1 ) + ( n r − 2 ) = [ ( n r ) + ( n r − 1 ) ] [ ( n r − 1 ) + ( n r − 2 ) ] = ( n + 1 r ) + ( n + 1 r − 1 ) = ( n + 2 r ) [ ∵ n C r + n C r − 1 = n + 1 C r ]

For 2≤r≤n, (nr)+2(nr−1)+(nr−2) is equal to

Question

For

2 \leq r \leq n, (\begin{matrix} n r \end{matrix}) + 2 (\begin{matrix} n r - 1 \end{matrix}) + (\begin{matrix} n r - 2 \end{matrix})

is equal to

Options:

A . (n+1r−1)

B . (n+1r+1)

C . 2(n+2r)

D . (n+2r)

Answer: Option D
:
D

(\begin{matrix} n r \end{matrix}) + 2 (\begin{matrix} n r - 1 \end{matrix}) + (\begin{matrix} n r - 2 \end{matrix}) = [(\begin{matrix} n r \end{matrix}) + (\begin{matrix} n r - 1 \end{matrix})]

[(\begin{matrix} n r - 1 \end{matrix}) + (\begin{matrix} n r - 2 \end{matrix})] = (\begin{matrix} n + 1 r \end{matrix}) + (\begin{matrix} n + 1 r - 1 \end{matrix}) = (\begin{matrix} n + 2 r \end{matrix})

[∵^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}]

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