$\sum\limits_{k = 0}^{10} {^{20}{C_k}} $ अर्थात् $^{20}{C_0} + {\,^{20}}{C_1} + …… + {\,^{20}}{C_{10}}$

हम जानते हैं कि ${(1 + x)^n} = {\,^n}{C_0} + {\,^n}{C_1}{x^1} + {\,^n}{C_2}{x^2} + …. + {\,^n}{C_n}.{x^n}$

$x = 1$ रखने पर, ${2^n} = {\,^n}{C_0} + {\,^n}{C_1} + {\,^n}{C_2} + ….. + {\,^n}{C_n}$

$n = 20$ रखने पर, ${2^{20}} = {\,^{20}}{C_0} + {\,^{20}}{C_1} + {\,^{20}}{C_2} + …… + {\,^{20}}{C_{20}}$

${2^{20}} + \,{\,^{20}}{C_{10}} = 2\,[{\,^{20}}{C_0} + {\,^{20}}{C_1} + …… + {\,^{20}}{C_{10}}]$

${[^{20}}{C_0} + {\,^{20}}{C_1} + …… + {\,^{20}}{C_{10}}] = {2^{19}} + \frac{1}{2}{\,^{20}}{C_{10}}$

$\sum\limits_{k = 0}^{10} {^{20}{C_k}}  = {2^{19}} + \frac{1}{2}{\,^{20}}{C_{10}}$.