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Question
Find the number of terms in the expansion of\[\left( a + b + c \right)^n\]
Solution
We have,
\[ = a^n +^n C_1 a^{n - 1} \left( b + c \right)^1 +^n C_2 a^{n - 2} \left( b + c \right)^2 + . . . +^n C_n \left( b + c \right)^n\]
First term consists of 1 term.
Second term on simplification gives 2 terms.
Third term on expansion gives 3 terms.
Similarly, fourth term on expansion gives 4 terms and so on.
∴ The total number of terms = 1 + 2 + 3 + .... + (n + 1) = \[\frac{\left( n + 1 \right)\left( n + 2 \right)}{2}\]
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