If some three consecutive in the binomial exp | vedclass

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Question

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

Vedclass · Accepted Answer

$232$

Vedclass · Answer

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 $=\frac{^{16} \mathrm{C}_{1}+^{16} \mathrm{C}_{2}+^{16} \mathrm{C}_{3}}{3}$

$=\frac{16+120+560}{3}$

$=232$

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

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