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Q.
The number of terms in the expansion of $(a + b + c)^n$, where $n \in N$ is
Binomial Theorem
Solution:
We have, $\left( a + b + c\right)^{n} = \left[ a + \left(b + c\right)\right]^{n}$
$= a^{n}+\,{}^{n}C_{1}\,a^{n-1}\left(b+c\right)^{1}+\,{}^{n}C_{2}\,a^{n-2}\left(b+c\right)^{2}+...$
$... + \,{}^{n}C_{n}\left(b+c\right)^{n}$
Further, expanding each term of R.H.S., we note that first term consist of $1$ term. Second term on simplification gives $2$ terms. Third term on simplification gives $3$ terms and so on.
$\therefore $ The total number of terms $= 1 + 2 + 3 + ... + \left(n + 1\right)$
$= \frac{\left(n+1\right)\left(n+2\right)}{2}$