Solve the Following Questions

Question:

$\sum_{k=0}^{20}\left({ }^{20} \mathrm{C}_{k}\right)^{2}$ is equal to:

  1. ${ }^{40} \mathrm{C}_{21}$

  2. ${ }^{40} \mathrm{C}_{19}$

  3. ${ }^{40} \mathrm{C}_{20}$

  4. ${ }^{41} \mathrm{C}_{20}$


Correct Option: , 3

Solution:

$\sum_{\mathrm{k}=0}^{20}{ }^{20} \mathrm{C}_{\mathrm{k}} \cdot{ }^{20} \mathrm{C}_{20-\mathrm{k}}$

sum of suffix is const. so summation will be ${ }^{40} \mathrm{C}_{20}$

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