If the coefficient of x^r in the expansion of | vedclass

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(C) Let $f(x) = (1+x+x^2+x^3)^{100} = \sum_{r=0}^{300} a_r x^r$. Setting $x=1$,we get $S = \sum_{r=0}^{300} a_r = f(1) = (1+1+1^2+1^3)^{100} = 4^{100}

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$(150) S$

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(C) Let $f(x) = (1+x+x^2+x^3)^{100} = \sum_{r=0}^{300} a_r x^r$. Setting $x=1$,we get $S = \sum_{r=0}^{300} a_r = f(1) = (1+1+1^2+1^3)^{100} = 4^{100}$. Differentiating $f(x)$ with respect to $x$: $f'(x) = 100(1+x+x^2+x^3)^{99} \cdot (1+2x+3x^2) = \sum_{r=1}^{300} r \cdot a_r x^{r-1}$. Setting $x=1$: $\sum_{r=0}^{300} r \cdot a_r = f'(1) = 100(4^{99}) \cdot (1+2+3) = 100 \cdot 4^{99} \cdot 6 = 600 \cdot 4^{99}$. Since $S = 4^{100}$,we have $4^{99} = \frac{S}{4}$. Therefore,$\sum_{r=0}^{300} r \cdot a_r = 600 \cdot \frac{S}{4} = 150 S$.

If the coefficient of $x^r$ in the expansion of $(1+x+x^2+x^3)^{100}$ is $a_r$,and $S = \sum_{r=0}^{300} a_r$,then $\sum_{r=0}^{300} r \cdot a_r =$

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