Question

In the expansion of ${(1+x)}^n$, The binomial coefficients of three consecutive terms are respectively 220, 495 and 795, the value

• A. $10$
• B. $11$
• C. $12$
• D. $13$

Answer 1

The correct option is C $12$

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

Coefficient of these terms are ${}^n{C}_{r-2},{}^n{C}_{r-1}$ and ${}^n{C}_r$ respectively.

These coefficient are equal to $220,495$ and $792$

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

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

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

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

Also,

$\frac{{}^n{C}_r}{{}^n{C}_{r-1}}=\frac{792}{495}$

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

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

$\Rightarrow$ $5n+5=13r$

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

$\Rightarrow$ $n=12$