In the expansion of
Question:

In the expansion of (1 + x)n the binomial coefficients of three consecutive terms are respectively 220, 495 and 792, find the value of n.

Solution:

Suppose the three consecutive terms are $T_{r-1}, T_{r}$ and $T_{r+1}$.

Coefficients of these terms are ${ }^{n} C_{r-2},{ }^{n} C_{r-1}$ and ${ }^{n} C_{r}$, respectively.

These coefficients are equal to 220,495 and 792 .

$\therefore \frac{{ }^{n} C_{r-2}}{{ }^{n} C_{r-1}}=\frac{220}{495}$

$\Rightarrow \frac{r-1}{n-r+2}=\frac{4}{9}$

$\Rightarrow 9 r-9=4 n-4 r+8$

$\Rightarrow 4 n+17=13 r \quad \ldots(1)$

Also,

$\frac{{ }^{n} C_{r}}{{ }^{n} C_{r-1}}=\frac{792}{495}$

$\Rightarrow \frac{n-r+1}{r}=\frac{8}{5}$

$\Rightarrow 5 n-5 r+5=8 r$

$\Rightarrow 5 n+5=13 r$

$\Rightarrow 5 n+5=4 n+17 \quad[$ From Eqn (1) $]$

$\Rightarrow n=12$

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