JEE Main & Advanced Mathematics Binomial Theorem and Mathematical Induction Properties of Binomial Coefficients

In the binomial expansion of \[{{(1+x)}^{n}},\]

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

where \[^{n}{{C}_{0}},{{\,}^{n}}{{C}_{1}},{{\,}^{n}}{{C}_{2}},......,{{\,}^{n}}{{C}_{n}}\] are the coefficients of various powers of \[x\] and called binomial coefficients, and they are written as \[{{C}_{0}},\,{{C}_{1}},\,{{C}_{2}},\,.....{{C}_{n}}\].

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

(1) The sum of binomial coefficients in the expansion of \[{{(1+x)}^{n}}\] is \[{{2}^{n}}\].

Putting \[x=1\] in (i), we get \[{{2}^{n}}={{C}_{0}}+{{C}_{1}}+{{C}_{2}}+.....+{{C}_{n}}\]      .....(ii)

(2) Sum of binomial coefficients with alternate signs : Putting \[x=-1\] in (i)

We get, \[{{C}_{0}}-{{C}_{1}}+{{C}_{2}}-{{C}_{3}}+......=0\]                        …..(iii)

(3) Sum of the coefficients of the odd terms in the expansion of \[{{(1+x)}^{n}}\] is equal to sum of the coefficients of even terms and each is equal to \[{{2}^{n-1}}\].

(4) \[^{n}{{C}_{r}}=\frac{n}{r}{{\,}^{n-1}}{{C}_{r-1}}=\frac{n}{r}\,.\frac{n-1}{r-1}{{\,}^{n-2}}{{C}_{r-2}}\] and so on.

(5) Sum of product of coefficients in the expansion is \[^{2n}{{C}_{n+r}}\].

Hence,\[{{C}_{0}}{{C}_{r}}+{{C}_{1}}{{C}_{r+1}}+......{{C}_{n-r}}.{{C}_{n}}{{=}^{2n}}{{C}_{n+r}}\]= \[\frac{2n!}{(n-r)!\text{ }(n+r)!}\]                .....(iv)

(6) Sum of squares of coefficients : Putting \[{{(a+b)}^{m}}={{a}^{m}}+m{{a}^{m-1}}b\] in (iv), we get  \[^{2n}{{C}_{n}}=C_{0}^{2}+C_{1}^{2}+......+C_{n}^{2}\]

(7) \[^{n}{{C}_{r}}{{+}^{n}}{{C}_{r-1}}{{=}^{n+1}}{{C}_{r}}\]

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

(9) \[{{C}_{1}}-2{{C}_{2}}+3{{C}_{3}}-................=0\]

(10) \[{{C}_{0}}+2{{C}_{1}}+3{{C}_{2}}+........+(n+1){{C}_{n}}=(n+2){{2}^{n-1}}\]

(11) \[C_{0}^{2}+C_{1}^{2}+C_{2}^{2}+...........+C_{n}^{2}=\frac{(2n)\,!}{{{(n\,!)}^{2}}}\]

(12) \[C_{0}^{2}-C_{1}^{2}+C_{2}^{2}-C_{3}^{2}+...........=\left\{ \begin{matrix} 0,\,\,\text{if }n\text{ is odd}\,\,\,\,\,\,\,\,\,\,\,  \\ {{(-1)}^{n/2}}{{.}^{n}}{{C}_{n/2}},\,\text{if }n\,\text{is even}  \\ \end{matrix} \right.\]