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$$\sum_{i=0}^{10}\sum_{i=j}^{10}{}^{10}C_j {}^{j}C_i $$ I tried to expand the terms ,but I can't put sum them to get a legit answer.

${}^{10}C_0 {}^{0}C_0+{}^{10}C_1 {}^{1}C_0.... +{}^{10}C_1 {}^{1}C_1 +{}^{10}C_2 {}^{2}C_1+.......... +{}^{10}C_2 {}^{2}C_2+{}^{10}C_3 {}^{3}C_2 +....+{}^{10}C_{10}+ {}^{10}C_{10} $

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$$\begin{align} \sum_{i=0}^{10}\sum_{j=i}^{10}\binom {10}j\binom ji &=\sum_{0\le i\le j\le 10}\binom {10}j\binom ji\\ &=\sum_{j=0}^{10}\sum_{i=0}^{j}\binom {10}j\binom ji\\ &=\sum_{j=0}^{10}\binom {10}j\sum_{i=0}^{j}\binom ji\\ &=\sum_{j=0}^{10}\binom {10}j2^j\\ &=(1+2)^{10}\\ &=3^{10} \end{align}$$

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    can you explain it a little. – 2017-01-17
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    @Boris - have added two lines. hope this makes it clearer. – 2017-01-17
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    NICE hypergeometric, HAVE ANY IDEA HOW TO SOLVE IT USING COMBINATIONAL ARGUMENT, tHANKS – 2017-01-17