Question

If the number of terms in the expansion of $(x+y+z{)}^n$ is $231$, then the value of $n$ is

• A. $20$
• B. $30$
• C. $31$
• D. $21$

Answer 1

The correct option is A $20$

Given expansion $(x+y+z{)}^n$
The number of terms
$231={}^{n+3-1}{C}_{3-1}\Rightarrow 231={}^{n+2}{C}_2\Rightarrow \frac{(n+2)(n+1)}{2}=231\Rightarrow n^2+3n+2=462\Rightarrow n^2+3n-460=0\Rightarrow (n+23)(n-20)=0\Rightarrow n=20(\because n\in N)$