# The number of terms in the expansion of $(x+y+z)^n$ ________

$\begin{array}{1 1}(A)\;\large\frac{n+1}{2}\\(B)\;\large\frac{n+2}{2}\\(C)\;(\large\frac{n+1}{2})(\large\frac{n+2}{2})\\(D)\;\text{None of these}\end{array}$

Toolbox:
• $(x+a)^n=nC_0x^n a^0+nC_1x^{n-1}a^1+nC_2 x^{n-2} a^2+nC_nx^0 a^n$
No. of terms in $(x+y+z)^n$
$\Rightarrow [x+(y+z)]^n$
$\Rightarrow [x+(y+z)]^n=nC_0x^n (y+z)^0+nC_1 x^{n-1}(y+z)^1+nC_2x^{n-2}(y+z)^2+.....nC_n x^0(y+z)^n$
$\therefore$ No. of terms in $(x+y+z)^n$
$\Rightarrow \large\frac{n+1}{2}[\frac{n+2}{2}]$
$\Rightarrow (\large\frac{n+1}{2})(\frac{n+2}{2})$
Hence (C) is the correct answer.