Question

If $n$ is an 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

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

Susbtitute in terms of n to get $={\left(\frac{n}{2}+1\right)}^2={\left(\frac{n+2}{2}\right)}^2$