With usual notations the value of n0+n+11+n+22+…+n+rr=

C0 n+C1 n+1+C2 n+2+………+Cr n+r =Cn n+Cn n+1+Cn n+2+…………+Cn n+r =Coefficient of xnin 1+xn+1+xn+1+1+xn+2+……+1+xn+r =Coeff of xn in 1+xn1+xr+1-1x = Coeff of xn+1 in 1+xn+r+1 -coeff of xn+1 in 1+xn = Cn+1 n+r+1-0=Cr n+r+1

With usual notations the value of n0+n+11+n+22+…+n+rr=

C0 n+C1 n+1+C2 n+2+………+Cr n+r =Cn n+Cn n+1+Cn n+2+…………+Cn n+r =Coefficient of xnin 1+xn+1+xn+1+1+xn+2+……+1+xn+r =Coeff of xn in 1+xn1+xr+1-1x = Coeff of xn+1 in 1+xn+r+1 -coeff of xn+1 in 1+xn = Cn+1 n+r+1-0=Cr n+r+1

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With usual notations the value of $(\begin{matrix} n \\ 0 \end{matrix}) + (\begin{matrix} n + 1 \\ 1 \end{matrix}) + (\begin{matrix} n + 2 \\ 2 \end{matrix}) + \dots + (\begin{matrix} n + r \\ r \end{matrix}) =$

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${}^{n + r + 1}C_{r}$

${}^{2 n + r}C_{n + 1}$

${}^{n}C_{r}$

${}^{n + r}C_{r + 1}$

answer is A.

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Detailed Solution

${}^{n}C_{0} + {}^{n + 1}C_{1} + {}^{n + 2}C_{2} + \dots \dots \dots + {}^{n + r}C_{r} = {}^{n}C_{n} + {}^{n + 1}C_{n} + {}^{n + 2}C_{n} + \dots \dots \dots \dots + {}^{n + r}C_{n} = Coefficient of x^{n} in [{(1 + x)}^{n} + {(1 + x)}^{n + 1} + {(1 + x)}^{n + 2} + \dots \dots + {(1 + x)}^{n + r}] = Coeff of x^{n} in [{(1 + x)}^{n} \frac{[{(1 + x)}^{r + 1} - 1]}{x}] = Coeff of x^{n + 1} in {(1 + x)}^{n + r + 1} - coeff of x^{n + 1} in {(1 + x)}^{n} = {}^{n + r + 1}C_{n + 1} - 0 = {}^{n + r + 1}C_{r}$