The value of 21 C 1 - 10 C 1 + 21 C 2 - 10 C 2 + 21 C 3 - 10 C 3 + 21 C 4 - 10 C 4 + … + 21 C 10 - 10…

We have,

$\left({}^{21}{C}_1{-}^{10}{C}_1\right)+\left({}^{21}{C}_2{-}^{10}{C}_2\right)+\left({}^{21}{C}_3{-}^{10}{C}_3\right)+\left({}^{21}{C}_4{-}^{10}{C}_4\right)+\dots +\left({}^{21}{C}_{10}{-}^{10}{C}_{10}\right)$

$=\left({}^{21}{C}_1+{ }^{21}{C}_2+{ }^{21}{C}_3+\dots +{ }^{21}{C}_{10}\right)-\left({}^{10}{C}_1{+}^{10}{C}_2{+}^{10}{C}_3+\dots {+}^{10}{C}_{10}\right)$

$=\frac{1}{2}\left[{}^{21}{C}_1{+}^{21}{C}_2{+}^{21}{C}_3+\dots {+}^{21}{C}_{10}{+}^{21}{C}_{11}+\dots {+}^{21}{C}_{20}\right]-\left(2^{10}-1\right)$

$=\frac{1}{2}\left(2^{21}-2\right)-\left(2^{10}-1\right)$

$=2^{20}-1-2^{10}+1$

$=2^{20}-2^{10}$.