Question

If $n$ is even integer and $a,b,c$ are distinct, the number of distinct terms in the expansion of ${(a+b+c)}^n+{(a+b-c)}^n$ is

• A. ${\left(\frac{n}{2}\right)}^2$
• B. ${\left(\frac{n+1}{2}\right)}^2$
• C. ${\left(\frac{n+2}{2}\right)}^2$
• D. ${\left(\frac{n+3}{2}\right)}^2$

Answer 1

The correct option is C ${\left(\frac{n+2}{2}\right)}^2$

Let $n=2m,m\in N$

$\therefore {(a+b+c)}^n+{(a+b-c)}^n={[(a+b)+c]}^{2m}+{[(a+b)-c]}^{2m}=2\{{(a+b)}^{2m}{+}^{2m}{C}_2{(a+b)}^{2m-2}{c}^2+...{+}^{2m}{C}_{2m}{c}^{2m}\}$

Therefore, the number of distinct terms in the expansion

$=(2m+1)+(2m-1)+...+3+1=\left(\frac{m+1}{1}\right).(2m+1+1)$

$={(m+1)}^2={(1+\frac{n}{2})}^2={\left(\frac{n+2}{2}\right)}^2$