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 $(1+x{)}^n$

$(1+x{)}^n$ = ${}^n{C}_0$ + ${}^n{C}_1$x + ${}^n{C}_2$$x^2$ ...............${}^n{C}_c$$x^n$

When we put x = 1, we get $S_1$ and when we put x = -1, we get $S_2$

⇒ $S_1$ = ${}^n{C}_0$ + ${}^n{C}_1$ + ${}^n{C}_2$ ........${}^n{C}_n$ = $2^n$

$S_2$ = ${}^n{C}_0$ - ${}^n{C}_1$ + ${}^n{C}_2$ ........= 0

⇒ $\frac{S_1}{S_1+S_2}$ = $\frac{S_1}{S_1}$ = 1