Question

The sum of last ten coeffficients in the expansion of $(1+x{)}^{19}$ when expanded in ascending powers of $x$ is

• A. $2^{18}$
• B. $2^{19}$
• C. $2^{18}-$${}^{19}{\text{C}}_{10}$
• D. none of these

Answer 1

The correct option is A $2^{18}$

The last ten coefficients are

${}^{19}{C}_{19},{}^{19}{C}_{18}...{}^{19}{C}_{11},{}^{19}{C}_{10}$

Hence

$S={}^{19}{C}_{19}+{}^{19}{C}_{18}+...{}^{19}{C}_{10}$.

Now we know, that sum of all coefficients is $2^{19}$, that is

${}^{19}{C}_0+{}^{19}{C}_1+...{}^{19}{C}_{18}+{}^{19}{C}_{19}=2^{19}$.

Also, ${}^n{C}_r={}^n{C}_{n-r}$

Hence

${}^{19}{C}_0+{}^{19}{C}_1...+{}^{19}{C}_9+{}^{19}{C}_{10}+...{}^{19}{C}_{18}+{}^{19}{C}_{19}=2^{19}$

$2({}^{19}{C}_{10}+{}^{19}{C}_{11}+...{}^{19}{C}_{18}+{}^{19}{C}_{19})=2^{19}$

$2S=2^{19}$

$S=2^{18}$.