$\begin{array}{l}
a){11^{10}} - 1 = {\left( {1 + 10} \right)^{10}} - 1 = \sum\limits_{k = 0}^{10} {C_{10}^k{{10}^k}} - 1\\
= C_{10}^0{.10^0} + C_{10}^1{.10^1} + C_{10}^2{.10^2} + ... + C_{10}^{10}{.10^{10}} - 1\\
= 1 + 100 + C_{10}^2{.10^2} + ... + C_{10}^{10}{.10^{10}} - 1\\
= 100 + C_{10}^2{.10^2} + ... + C_{10}^{10}{.10^{10}} \vdots 100\left( {dpcm} \right)\\
b){101^{100}} - 1 = {\left( {1 + 100} \right)^{100}} - 1 = \sum\limits_{k = 0}^{100} {C_{100}^k{{100}^k}} - 1\\
= C_{100}^0{.100^0} + C_{100}^1{.100^1} + C_{100}^2{.100^2} + ... + + C_{100}^{100}{.100^{100}} - 1\\
= 1 + 100.100 + C_{100}^2{.100^2} + ... + + C_{100}^{100}{.100^{100}} - 1\\
= 10000 + C_{100}^2{.100^2} + ... + + C_{100}^{100}{.100^{100}} \vdots 10000\left( {dpcm} \right)\\
c)\sqrt {10} \left[ {{{\left( {1 + \sqrt {10} } \right)}^{100}} - {{\left( {1 - \sqrt {10} } \right)}^{100}}} \right]\\
= \sqrt {10} \left[ {\left( {1 + C_{100}^1.\sqrt {10} + C_{100}^2.{{\sqrt {10} }^2} + ... + C_{100}^{100}.{{\sqrt {10} }^{100}}} \right) - \left( {1 - C_{100}^1.\sqrt {10} + C_{100}^2.{{\sqrt {10} }^2} - ... + C_{100}^{100}.{{\sqrt {10} }^{100}}} \right)} \right]\\
= \sqrt {10} \left[ {2C_{100}^1.\sqrt {10} + 2C_{100}^3.{{\sqrt {10} }^3} + ... + 2C_{100}^{99}.{{\sqrt {10} }^{99}}} \right]\\
= 2C_{100}^1.10 + 2C_{100}^3{.10^2} + ... + 2C_{100}^{99}{.10^{50}}\\
la\,mot\,so\,nguyen \Rightarrow dpcm
\end{array}$
