Question

In the expansion $(1+x{)}^n$ if the coefficient of the 8th term is 4 times the coefficient of the 9th term, then find the total number of terms in the expansion.

• A. 9
• B. 10
• C. 8
• D. none of these

Answer 1

The correct option is A 9

Given, $(1+x{)}^n$

General term,

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

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

$T_8=T_{7+1}={}^n{C}_7\left(1{)}^{n-7}\right(x{)}^7$

$T_9=T_{8+1}={}^n{C}_8\left(1{)}^{n-8}\right(x{)}^8$

Given, the coefficient of $8$th term is $4$ times coefficient of $9$th term

${}^n{C}_7=4{}^n{C}_8$

$\frac{n!}{7!(n-7)!}=4\times \frac{n!}{8!(n-8)!}$ ..................... $n!=n(n-1)!$

$8!(n-8)!=4\times 7!(n-7)!$

$8\times 7!(n-8)!=4\times 7!\times (n-7)(n-8)!$

$8=4(n-7)$

$2=n-7$

$n=9$

Hence, the total number of terms in the expansion are $9$

Hence, correct option is A.