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)

232

Answer 2

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