JEE Main & Advanced Mathematics Binomial Theorem and Mathematical Induction Some Important Expansions

(1) Replacing\[y\]y  by \[-y\]   in (i), we get,

\[{{(x-y)}^{n}}={{\,}^{n}}{{C}_{0}}{{x}^{n-0}}{{y}^{0}}{{-}^{n}}{{C}_{1}}{{x}^{n-1}}{{y}^{1}}{{+}^{n}}{{C}_{2}}{{x}^{n-2}}{{y}^{2}}-.......\]\[+{{(-1)}^{r}}{{\,}^{n}}{{C}_{r}}{{x}^{n-r}}{{y}^{r}}+....+{{(-1)}^{n}}{{\,}^{n}}{{C}_{n}}{{x}^{0}}{{y}^{n}}\]

i.e., \[{{(x-y)}^{n}}=\sum\limits_{r=0}^{n}{{{(-1)}^{r}}{{\,}^{n}}{{C}_{r}}{{x}^{n-r}}{{y}^{r}}}\]                           

The terms in the expansion of \[{{(x-y)}^{n}}\] are alternatively positive and negative, the last term is positive or negative according as \[n\] is even or odd.

(2) Replacing \[x\] by 1 and y by \[x\] in equation (i) we get,  \[{{(1+x)}^{n}}{{=}^{n}}{{C}_{0}}{{x}^{0}}{{+}^{n}}{{C}_{1}}{{x}^{1}}{{+}^{n}}{{C}_{2}}{{x}^{2}}+......+{{\,}^{n}}{{C}_{r}}{{x}^{r}}+......{{+}^{n}}{{C}_{n}}{{x}^{n}}\]

i.e., \[{{(1+x)}^{n}}=\sum\limits_{r=0}^{n}{^{n}{{C}_{r}}{{x}^{r}}}\]

This is expansion of \[{{(1+x)}^{n}}\] in ascending power of \[x\].

(3) Replacing \[x\] by 1 and \[y\] by \[-x\] in (i) we get, 

\[{{(1-x)}^{n}}={{\,}^{n}}{{C}_{0}}{{x}^{0}}-{{\,}^{n}}{{C}_{1}}{{x}^{1}}+{{\,}^{n}}{{C}_{2}}{{x}^{2}}-.....\]  \[+{{(-1)}^{r}}{{\,}^{n}}{{C}_{r}}{{x}^{r}}+....+{{(-1)}^{n}}{{\,}^{n}}{{C}_{n}}{{x}^{n}}\]

 i.e.,  \[{{(1-x)}^{n}}=\sum\limits_{r=0}^{n}{{{(-1)}^{r}}{{\,}^{n}}{{C}_{r}}{{x}^{r}}}\]

(4) \[{{(x+y)}^{n}}+{{(x-y)}^{n}}=\]\[2\,{{[}^{n}}{{C}_{0}}{{x}^{n}}{{y}^{0}}{{+}^{n}}{{C}_{2}}{{x}^{n-2}}{{y}^{2}}\]\[{{+}^{n}}{{C}_{4}}{{x}^{n-4}}{{y}^{4}}+.......]\] and

\[{{(x+y)}^{n}}-{{(x-y)}^{n}}=2\,{{[}^{n}}{{C}_{1}}{{x}^{n-1}}{{y}^{1}}\]\[{{+}^{n}}{{C}_{3}}{{x}^{n-3}}{{y}^{3}}+{{\,}^{n}}{{C}_{5}}{{x}^{n-5}}{{y}^{5}}+...]\]

(5) The coefficient of \[{{(r+1)}^{th}}\] term in the expansion of \[{{(1+x)}^{n}}\] is \[^{n}{{C}_{r}}\].

(6) The coefficient of \[{{x}^{r}}\] in the expansion of \[{{(1+x)}^{n}}\] is \[^{n}{{C}_{r}}\].