# The number of terms in the expansion of $(a+b+c)^{n}$ is Option 1) $\frac{(n+1)\: (n+2)}{2}$ Option 2) $n+3$ Option 3) $\frac{n(n+1)}{2}$ Option 4) None of these

D Divya Saini

As learnt in concept

Theorem of Combinations -

The number of combinations of n distinct objects taken r at a time when any object may be repeated any number of times is $^{n+r-1}c_{r}$.

- wherein

Coefficient of $x^{r}$ in $(1-x)^{-n}$.

$(a+b+c)^{n}$

The general term is $_{C_{r}}^{n}a^{x}b^{y}C^{z}$

Sum of powers =n

x+y+z=n

Thus no of solutions = $^{n+2}{C_{2}}$

Option 1)

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

This is correct option

Option 2)

$n+3$

This is incorrect option

Option 3)

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

This is incorrect option

Option 4)

None of these

This is incorrect option

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