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  • <img alt="\text { The sum } 2 \times{ }^{40} C_2+6 \times{ }^{40} C_3+12 \times{ }^{40} C_4+20 \times{ }^{40} C_5+\ldots+1560" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Ctext%20%7B%20T

\text { The sum } 2 \times{ }^{40} C_2+6 \times{ }^{40} C_3+12 \times{ }^{40} C_4+20 \times{ }^{40} C_5+\ldots+1560 \times{ }^{40} C_{40} is not divisible by

Option: 1

3


Option: 2

5


Option: 3

13


Option: 4

2^{42}


Answers (1)

best_answer

2 \times{ }^{40} C_2+6 \times{ }^{40} C_3+12 \times{ }^{40} C_4+20 \times{ }^{40} C_5+\ldots+1560 \times{ }^{40} C_{40}=\sum_{r=2}^{40}\left(r^2-r\right) \cdot{ }^{40} C_r

\begin{aligned} & =\sum_{r=2}^{40} r(r-1) \cdot{ }^{40} C_r \\ & =\sum_{r=2}^{40} 40(r-1) \cdot{ }^{39} C_{r-1} \\ & =\sum_{r=2}^{40} 40 \times 39 \cdot{ }^{38} C_{r-2} \\ & =40 \times 39 \times 2{ }^{38} \\ & =5 \times 39 \times 2^{41} \end{aligned}

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himanshu.meshram

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