# The number of terms in the expansion of (x + y + z)

Question:

The number of terms in the expansion of (x + y + z)n is ___________.

Solution:

$(x+y+z)^{n}$

$=[x+(y+z)]^{n}$

$={ }^{n} C_{0} x^{n}+{ }^{n} C_{1} x^{n-1}(y+z)+{ }^{n} C_{2} x^{n-2}(y+z)^{2}+\ldots+{ }^{n} C_{n}(y+z)^{n}$

∴ Number of terms in the expansion

$=1+2+\ldots+n+(n+1)$

$=\frac{(n+1)(n+2)}{2}={ }^{n+2} C_{2}$