Question

Let $S_1$ = ${}^n{C}_0$ + ${}^n{C}_1$ + ${}^n{C}_2$.............${}^n{C}_n$ and $S_2$ = ${}^n{C}_0$ - ${}^n{C}_1$ + ${}^n{C}_2$ ..............+ $(-1{)}^n$ ${}^n{C}_n$

Find the value of $\frac{S_1}{S_1+S_2}$ is ___.

Answer 1

Consider the expansion of $(1+x{)}^n$

${\left(1+x\right)}^n=nC0+nC1x+nC2x^2+....+nCn{x}^n$

When we put x = 1, we get $S_1$ and when we put x = -1, we get $S_2$
$S_1=nC0+nC1+nC2+....+nCn=2^n$
$S_2=nC0-nC1+nC2+....=0$
$\frac{S_1}{S_1+S_2}=\frac{S_1}{S_1}=1$