Question

If the coefficients of x in the 4th and 7th terms of the expansion $(1+x{)}^n$ are in the ratio 4:7, then find n

• A. 8
• B. 9
• C. 12
• D. 10

Answer 1

The correct option is D 10

Given,

$(1+x{)}^n$

General term $T_{r+1}={}^n{C}_r{r}^{n-r}{y}^r$

$T_{r+1}={}^n{C}_r(1{)}^{n-r}{x}^r$

Given, the coefficients of x in the $4$th and $7$th terms are in ratio $4:7$

$T_4=T_{3+1}={}^n{c}_3(1{)}^{n-3}{x}^3$

$T_7=T_{6+1}={}^n{C}_6\left(1{)}^{n-6}\right(x{)}^6$

$\frac{4}{7}=\frac{{}^n{C}_3}{{}^n{C}_6}$

$\frac{4}{7}=\frac{n!6!(n-6)!}{3!(n-3)!n!}$

$\frac{4}{7}=\frac{6!(n-6)!}{3!(n-3)(n-4)(n-5)(n-6)!}$

$\frac{4}{7}=\frac{4\times 5\times 6}{(n-3)(n-4)(n-5)}$

$(n-3)(n-4)(n-5)=210$

Checking through options

$n=10$.