Question

If some three consecutive in the binomial expansion of $(x+1{)}^n$ is powers of $x$ are in the ratio $2:15:70$, then the average of these three coefficient is :-

• A. 964
• B. 625
• C. 227
• D. 232

Answer 1

Answer: (D)

$\frac{{}^n{C}_{r-1}}{{}^n{C}_r}=\frac{2}{15}$
$\frac{\frac{n!}{\left(r-1\right)!\left(n-r+1\right)!}}{\frac{n!}{r!\left(n-r\right)!}}=\frac{2}{15}$
$\frac{r}{n-r+1}=\frac{2}{15}$
$15r=2n-2r+2$
$17r=2n+2$
$\frac{{}^n{C}_r}{{}^n{C}_{r+1}}=\frac{15}{70}$
$\frac{\frac{n!}{r!\left(n-r\right)!}}{\frac{n!}{\left(r+1\right)!\left(n-r-1\right)!}}=\frac{3}{14}$
$\frac{r+1}{n-r}=\frac{3}{14}$
$14r+14=3n-3r$
$3n-17r=14$
$\frac{2n-17r=-2}{n=16}$
$17r=34,r=2$
${}^{16}{C}_1,{}^{16}{C}_2,{}^{16}{C}_3$
$\frac{{}^{16}{C}_1{+}^{16}{C}_2{+}^{16}{C}_3}{3}=\frac{16+120+560}{3}$
$\frac{680+16}{3}=\frac{696}{3}=232$