- Home
- Standard 11
- Mathematics
$\sum\limits_{k = 0}^{10} {^{20}{C_k} = } $
${2^{19}} + \frac{1}{2}{\,^{20}}{C_{10}}$
${2^{19}}$
$^{20}{C_{10}}$
એકપણ નહિ.
Solution
(a) $\sum\limits_{K = 0}^{10} {^{20}{C_k}} $ i.e., $^{20}{C_0} + {\,^{20}}{C_1} + …… + {\,^{20}}{C_{10}}$
We know that, ${(1 + x)^n} = {\,^n}{C_0} + {\,^n}{C_1}{x^1} + {\,^n}{C_2}{x^2} + …. + {\,^n}{C_n}.{x^n}$
Put $x = 1$; ${2^n} = {\,^n}{C_0} + {\,^n}{C_1} + {\,^n}{C_2} + ….. + {\,^n}{C_n}$
Put $n = 20$; ${2^{20}} = {\,^{20}}{C_0} + {\,^{20}}{C_1} + {\,^{20}}{C_2} + …… + {\,^{20}}{C_{20}}$
${2^{20}} + \,{\,^{20}}{C_{10}} = 2\,[{\,^{20}}{C_0} + {\,^{20}}{C_1} + …… + {\,^{20}}{C_{10}}]$
${[^{20}}{C_0} + {\,^{20}}{C_1} + …… + {\,^{20}}{C_{10}}] = {2^{19}} + \frac{1}{2}{\,^{20}}{C_{10}}$
$\sum\limits_{k = 0}^{10} {^{20}{C_k}} = {2^{19}} + \frac{1}{2}{\,^{20}}{C_{10}}$.
