2 cdot {}^{20}C_0 + 5 cdot {}^{20}C_1 + 8 cdo | vedclass

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(D) The given series is $S = \sum_{r=0}^{20} (3r + 2) \cdot {}^{20}C_r$. $S = 3 \sum_{r=0}^{20} r \cdot {}^{20}C_r + 2 \sum_{r=0}^{20} {}^{20}C_r$. Us

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$2^{25}$

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(D) The given series is $S = \sum_{r=0}^{20} (3r + 2) \cdot {}^{20}C_r$. $S = 3 \sum_{r=0}^{20} r \cdot {}^{20}C_r + 2 \sum_{r=0}^{20} {}^{20}C_r$. Using the identity $r \cdot {}^{n}C_r = n \cdot {}^{n-1}C_{r-1}$,we get: $S = 3 \sum_{r=1}^{20} 20 \cdot {}^{19}C_{r-1} + 2 \cdot 2^{20}$. $S = 3 \cdot 20 \cdot \sum_{r=1}^{20} {}^{19}C_{r-1} + 2^{21}$. Since $\sum_{r=1}^{20} {}^{19}C_{r-1} = 2^{19}$,we have: $S = 60 \cdot 2^{19} + 2 \cdot 2^{20} = 30 \cdot 2^{20} + 2 \cdot 2^{20} = 32 \cdot 2^{20} = 2^5 \cdot 2^{20} = 2^{25}$.

$2 \cdot {}^{20}C_0 + 5 \cdot {}^{20}C_1 + 8 \cdot {}^{20}C_2 + 11 \cdot {}^{20}C_3 + \dots + 62 \cdot {}^{20}C_{20}$ is equal to

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