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Q.
$\displaystyle\sum_{0 \leq i \leq j \leq 10}\left({ }^{10} C_j\right)\left({ }^j C_i\right)$ is equal to
Binomial Theorem
Solution:
${ }^{10} C _0\left({ }^0 C _0\right)+{ }^{10} C _1\left({ }^1 C _0+{ }^1 C _1\right)+{ }^{10} C _2 \left({ }^2 C _0+{ }^2 C _1+{ }^2 C _2\right)+\ldots \ldots . . \\ +{ }^{10} C _{10}\left({ }^{10} C _0+{ }^{10} C _1+{ }^{10} C _2+\ldots \ldots .+{ }^{10} C _{10}\right)$
$={ }^{10} C _0+{ }^{10} C _1 \cdot 2+{ }^{10} C _2 \cdot 2^2+{ }^{10} C _3 \cdot 2^3+\ldots \ldots .+{ }^{10} C _{10} \cdot 2^{10}=(1+2)^{10}=3^{10}$