Number of Terms In The Expansion of \[{{(a+b+c)}^{n}}\] and \[{{(a+b+c+d)}^{n}}\]
Category : JEE Main & Advanced
\[{{(a+b+c)}^{n}}\]can be expanded as : \[{{C}_{0}}-{{C}_{1}}+{{C}_{2}}-{{C}_{3}}+......=0\]
\[={{(a+b)}^{n}}{{+}^{n}}{{C}_{1}}{{(a+b)}^{n-1}}{{(c)}^{1}}{{+}^{n}}{{C}_{2}}{{(a+b)}^{n-2}}{{(c)}^{2}}+.....+{{\,}^{n}}{{C}_{n}}\,{{c}^{n}}\]
\[=(n+1)\,\text{term }+n\,\text{term }+\text{ }(n-1)\text{term }+...+1\text{term}\]
\[\therefore \] Total number of terms = \[(n+1)+(n)+(n-1)+......+1=\frac{(n+1)(n+2)}{2}\].
Similarly, number of terms in the expansion of
\[{{(a+b+c+d)}^{n}}=\frac{(n+1)(n+2)(n+3)}{6}\].
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