Binomial Theorem for Positive Integral Index
Category : JEE Main & Advanced
The rule by which any power of binomial can be expanded is called the binomial theorem.
If \[n\] is a positive integer and \[x,\,\,y\,\,\in C\] then
\[{{(x+y)}^{n}}{{=}^{n}}{{C}_{0}}{{x}^{n-0}}{{y}^{0}}{{+}^{n}}{{C}_{1}}{{x}^{n-1}}{{y}^{1}}+{{\,}^{n}}{{C}_{2}}\,{{x}^{n-2}}{{y}^{2}}+........\]\[{{+}^{n}}{{C}_{r}}{{x}^{n-r}}{{y}^{r}}+......{{+}^{n}}{{C}_{n-1}}x{{y}^{n-1}}{{+}^{n}}{{C}_{n}}{{x}^{0}}{{y}^{n}}\]
i.e., \[{{(x+y)}^{n}}=\sum\limits_{r=0}^{n}{^{n}{{C}_{r}}.{{x}^{n-r}}.{{y}^{r}}}\] .....(i)
Here \[^{n}{{C}_{0}},\,{{}^{n}}{{C}_{1}},{{\,}^{n}}{{C}_{2}},{{......}^{n}}{{C}_{n}}\] are called binomial coefficients and \[{{(1+x)}^{n}}=1+nx+\frac{n(n-1)}{2!}{{x}^{2}}+.......\] for \[0\le r\le n\].
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