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  • Let  denotes the binomial coefficient of  in the expansion of . If <

Let ^nC_r denotes the binomial coefficient of x^r in the expansion of (1+x)^n. If \sum_{\mathrm{k}=0}^{10}\left(2^{2}+3 \mathrm{k}\right){ }^{\mathrm{10}} \mathrm{C}_{\mathrm{k}}=\alpha .3^{10}+\beta .2^{10}, \alpha, \beta \in \mathrm{R} then \alpha + \beta is equal to _________-
Option: 1 10
Option: 2 19
Option: 3 16
Option: 4 24

Answers (1)

best_answer

\\\sum_{\mathrm{k}=0}^{10} 4\left({ }^{\mathrm{n}} \mathrm{C}_{\mathrm{k}}\right)+3 \sum_{\mathrm{k}=0}^{10} \mathrm{k} \cdot{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{k}}\\=4\left({ }^{10} \mathrm{C}_{0}+{ }^{10} \mathrm{C}_{1}+\ldots . .+{ }^{10} \mathrm{C}_{10}\right)+3\left(1 .{ }^{10} \mathrm{C}_{1}+2 \cdot{ }^{10} \mathrm{C}_{2}+\ldots .+10 \cdot{ }^{10} \mathrm{C}_{9}\right)

\\=4\left(2^{10}\right)+3\left(10.2^{9}\right)\\=(4+15) 2^{10}=19 \cdot 2^{10} \\ \text { compare with } \alpha .3^{10}+\beta .2^{10} \\\Rightarrow \alpha=0, \beta=19 \Rightarrow \alpha+\beta=19

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himanshu.meshram

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