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7.Binomial Theorem
hard
If some three consecutive in the binomial expansion of ${\left( {x + 1} \right)^n}$ in 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
$232$
(JEE MAIN-2019)
Solution
Given: $\frac{^{n} C_{r-1}}{^{n} C_{r}}=\frac{2}{15}$
$\Rightarrow \frac{r}{n-r+1}=\frac{2}{15}$
$\Rightarrow 15 r=2 n-2 r+2$
$\Rightarrow 17 r=2 n+2………(1)$
also given $\frac{^{n} C_{r}}{^{n} C_{r+1}}=\frac{15}{70}$
$ \Rightarrow \frac{r+1}{n-r}=\frac{3}{14}$
$\Rightarrow 3 n-3 r=14 r+14………(2)$
$\Rightarrow 17 \mathrm{r}=3 \mathrm{n}-14$
Solving $(1)$ and $(2)$ $n=16, r=2$
Average of coefficient $=\frac{^{16} \mathrm{C}_{1}+^{16} \mathrm{C}_{2}+^{16} \mathrm{C}_{3}}{3}$
$=\frac{16+120+560}{3}$
$=232$
Standard 11
Mathematics
