JEE MainMathematicsToughBinomial Theorem

If [IMAGE],then the ordered pair (A, β) is equal to

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S = 1^{2}20C_{1} + 2^{2} \cdot \ ^{20}C_{2} + 3^{2}\ ^{20}C_{3} + \ldots\ldots\ldots\ldots\ldots\ldots + 20^{2}.\ ^{20}C_{20} = \sum_{r = 1}^{20}\mspace{2mu} r^{2} \cdot \ ^{20}C_{r}

= \sum_{r = 1}^{20}\mspace{2mu} r \cdot \left( r^{20}C_{r} \right) = 20\sum_{r = 1}^{20}\mspace{2mu} r \cdot \ ^{19}C_{r - 1}

= 20\sum_{r = 1}^{20}\mspace{2mu}(r - 1 + 1)\ ^{19}C_{r - 1} = 20\sum_{r = 1}^{20}\mspace{2mu}(r - 1)^{19}C_{r - 1} + 20\sum_{r = 1}^{20}\mspace{2mu}\ ^{19}C_{r - 1}

= 20 \times 19\sum_{r = 2}^{20}\mspace{2mu}\ ^{18}C_{r - 2} + 20 \times 2^{19}

= 20 \times 19 \times 2^{18} + 20 \times 2^{19} = 20 \times 2^{18}(19 + 2) = 20 \times 21 \times 2^{18}

= 420 \times 2^{18}

Source: Grass-G / M-28 Mathematics Paper FST 16 Jan.docx

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