Question

Find the value of ${}^{2n+1}{C}_0{+}^{2n+1}{C}_1{+}^{2n+1}{C}_2+......{+}^{2n+1}{C}_n{+}^{2n+1}{C}_{n+1}{+}^{2n+1}$$C_{n+2}+.....{+}^{2n+1}{C}_{2n+1}$ if $n=3$.
___

Answer 1

We know,
Sum of the coefficients of binomial expansion
${}^{2n+1}{C}_0+{}^{2n+1}{C}_1+{}^{2n+1}{C}_2+......+{}^{2n+1}{C}_n=2^n$
${}^{2n+1}{C}_0+{}^{2n+1}{C}_1+{}^{2n+1}{C}_2+......+{}^{2n+1}$$C_n+{}^{2n+1}{C}_{n+1}+{}^{2n+1}{C}_{n+2}+.....+{}^{2n+1}{C}_{2n+1}=2^{2n+1}$

Put n=3
$2^{6+1}=2^7=128$