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Multinomial Expansion

The number of...

Question

The number of terms in the expansion of $(x + y + z)^{n}$ is

\frac{n (n + 1)}{2}

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\frac{(n + 1) (n + 2)}{2}

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\frac{n (n + 3)}{2}

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\frac{(n + 1) (n + 3)}{2}

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Solution

The correct option is B $\frac{(n + 1) (n + 2)}{2}$
Number of terms is equal to $^{n + r - 1} C_{n}$
For $(x + y + z)^{n}$
$r = 3$
Hence number of terms are
$^{n + 3 - 1} C_{n}$
$=^{n + 2} C_{n}$
$= \frac{(n + 2)!}{2! n!}$
$= \frac{(n + 1) (n + 2)}{2}$

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