if n is an even integer and a, b, c are distinct numbers

then the number of distinct terms in the expansion of

$(a+b+c{)}^n+(a+b-c{)}^n,$ is

We have, 

$\begin{array}{l}(a+b+c{)}^n+(a+b-c{)}^n\\ =2\left\{{ }^n{c}_0(a+b{)}^n{c}^0{+}^n{c}_2(a+b{)}^{n-2}{c}^2+\dots ..\right\}\end{array}$

We observe that there are  $\left(\frac{n}{2}+1\right)$ distinct terms on RHS of the

above equality such that first term is the sum of (n + 1) terms,

 second term is the sum of  $(n -1)$  terms, third term is the sum of 

$(n- 3)$) terms and so on

   . Hence, required number of terms

 $=(n+1)+(n-1)+(n-3)+\dots .$ upto  $\left(\frac{n}{2}+1\right)$ terms

$\begin{array}{l}=\frac{1}{2}\left(\frac{n}{2}+1\right)\left\{2(n+1)+\left(\frac{n}{2}+1-1\right)\times (-2)\right\}\\ =\frac{n+2}{4}(n+2)={\left(\frac{n+2}{2}\right)}^2\end{array}$