Binomial Theorem for Positive Integral Index

If $\mathrm{x}$ and $\mathrm{y}$ are real, then for all $\mathrm{n} \in \mathrm{N}$

$ \begin{aligned} & (x+y)^{n}={ }^{n} C _{0} x^{n}+{ }^{n} C _{1} x^{n-1} y+{ }^{n} C _{2} x^{n-2} y^{2}+\ldots \ldots .+{ }^{n} C _{r} x^{n-r} y^{r}+\ldots .+{ }^{n} C _{n-1} x^{1} y^{n-1}+{ }^{n} C _{n} y^{n} \\ & =\quad \sum _{r=0}^{n}{ }^{n} C _{r} x^{n-r} y^{r} \\ & (x-y)^{n}={ }^{n} C _{0} x^{n}-{ }^{n} C _{1} x^{n-1} y^{1}+{ }^{n} C _{2} x^{n-2} y^{2} \ldots \ldots .{ }^{n} C _{r}(-1)^{r} x^{n-r} y^{r}+\ldots .{ }^{n} C _{n-1}(-1)^{n-1} x^{1} y^{n-1}+{ }^{n} C _{n}(-1)^{n} y^{n} \\ & (x+y)^{n}+(x-y)^{n}=2\left\{{ }^{n} C _{0} x^{n}+{ }^{n} C _{2} x^{n-2} y^{2}+{ }^{n} C _{4} x^{n-4} y^{4}+\ldots \ldots . .\right. \\ & (x+y)^{n}-(x-y)^{n}=2\left\{{ }^{n} C _{1} x^{n-1} a^{1}+{ }^{n} C _{3} x^{n-3} a^{3}+{ }^{n} C _{5} x^{n-5} y^{5}+\ldots \ldots . .\right] \\ & (1+x)^{n}={ }^{n} C _{0}+{ }^{n} C _{1} x+{ }^{n} C _{2} x^{2}+\ldots \ldots \ldots{ }^{n} C _{r} x^{r}+\ldots \ldots .{ }^{n} C _{n} x^{n} \\ & (1-x)^{n}={ }^{n} C _{0}-{ }^{n} C _{1} x+{ }^{n} C _{2} x^{2}-\ldots \ldots \ldots .{ }^{n} C _{r}(-1)^{r} x^{r}+\ldots \ldots+{ }^{n} C _{n}(-1)^{n} x^{n} \end{aligned} $

Properties of Binomial Expansion

(i). The number of terms in the expansion of $(x+y)^{n}$ where $n \in \mathrm{N}$ is $(\mathrm{n}+1)$.

(ii). The sum of exponents of $x & y$ in $(x+y)^{n}$ is equal to $n$, the index of the expansion.

(iii). Since ${ }^{n} \mathrm{C} _{\mathrm{r}}={ }^{\mathrm{n}} \mathrm{C} _{\mathrm{n}-\mathrm{r}} ; \mathrm{r}=0,1,2, \ldots \ldots . \mathrm{n}$, the binomial coefficients equidistant from the begin ning and the end are equal.

i.e. ${ }^{n} C _{0}={ }^{n} C _{n},{ }^{n} C _{1}={ }^{n} C _{n-1}$ and so on.

(iv). The general term is the expansion of $(\mathrm{x}+\mathrm{y})^{\mathrm{n}}$ is given by

$ T _{r+1}={ }^{n} C _{r} x^{n-r} y^{r} $

(v). Coefficient of $(r+1)^{\text {th }}$ term in the expansion of $(1+x)^{n}$ is ${ }^{n} C _{r}=$ coefficient of $x^{r}$.

(vi). If $\mathrm{n}$ is odd, then $\left\{(\mathrm{x}+\mathrm{y})^{\mathrm{n}}+(\mathrm{x}-\mathrm{y})^{\mathrm{n}}\right\}$ and $\left\{(\mathrm{x}+\mathrm{y})^{\mathrm{n}}-(\mathrm{x}-\mathrm{y})^{\mathrm{n}}\right\}$ have same number of terms equal to $\left(\frac{\mathrm{n}+1}{2}\right)$.

If $n$ is even, then

$\left\{(\mathrm{x}+\mathrm{y})^{\mathrm{n}}+(\mathrm{x}-\mathrm{y})^{\mathrm{n}}\right\}$ has $\left(\frac{\mathrm{n}}{2}+1\right)$ terms and

$\left\{(\mathrm{x}+\mathrm{y})^{\mathrm{n}}-(\mathrm{x}-\mathrm{y})^{\mathrm{n}}\right\}$ has $\left(\frac{\mathrm{n}}{2}\right)$ terms.

(vii). Middle term If $\mathrm{n}$ is even then in the expansion of $(x+y)^{n},\left(\frac{n}{2}+1\right)$ th terms is the middle term.

If $\mathrm{n}$ is odd natural number, then $\left(\frac{\mathrm{n}+1}{2}\right)$ th and $\left(\frac{\mathrm{n}+3}{2}\right)$ th are the middle terms in the expansion of $(x+y)^{n}$.

(viii). Let $\mathrm{S}=(\mathrm{x}+\mathrm{y})^{\mathrm{n}}={ }^{\mathrm{n}} \mathrm{C} _{0} \mathrm{x}^{\mathrm{n}}+{ }^{n} \mathrm{C} _{1} \mathrm{x}^{\mathrm{n}-1} \mathrm{y}+\ldots \ldots .+{ }^{\mathrm{n}} \mathrm{C} _{\mathrm{n}-1} \mathrm{xy}^{\mathrm{n}-1}+{ }^{n} \mathrm{C} _{\mathrm{n}} \mathrm{y}^{\mathrm{n}}$ where $\mathrm{n} \in \mathrm{N}$

$= \sum _{\mathrm{r}=0}^{\mathrm{n}} \mathrm{C} _{\mathrm{r}} \mathrm{x}^{\mathrm{n}-\mathrm{r}} \mathrm{y}^{\mathrm{r}}$

Replacing $\mathrm{r}$ by $\mathrm{n}-\mathrm{r}$ we get,

$S=\sum _{r=0}^{n}{ }^{n} C _{n-r} x^{r} y^{n-r}$

$={ }^{n} C _{n} y^{n}+{ }^{n} C _{n-1} y^{n-1} x+\ldots . .+{ }^{n} C _{n-1} y^{n-1}+{ }^{n} C _{n} x^{n}$.

i.e. By replacing $\mathrm{r}$ by $\mathrm{n}-\mathrm{r}$, we are writing the binomial expansion in the reverse order

Properties of Binomial Coefficient

(i). Sum of two binomial coefficients, ${ }^{n} C _{r}+{ }^{n} C _{r-1}={ }^{n+1} C _{r}$

(ii). ${ }^{n} C _{r}=\frac{n}{r}{ }^{n-1} C _{r-1}$

(iii). $\frac{{ }^{n} C _{r}}{{ }^{n} C _{r-1}}=\frac{n-r+1}{r}$

(iv). ${ }^{n} \mathrm{C} _{\mathrm{r}}={ }^{n} \mathrm{C} _{\mathrm{s}} \Rightarrow$ either $\mathrm{r}=\mathrm{s}$ or $\mathrm{r}+\mathrm{s}=\mathrm{n}$

Multinomial Theorem (For a positive integral index)

$\left(\mathrm{x} _{1}+\mathrm{x} _{2}+\ldots \ldots .+\mathrm{xk}\right)^{\mathrm{n}}=\frac{\mathrm{n} !}{\mathrm{n} _{1} ! \mathrm{n} _{2} ! \ldots . . \mathrm{nk} !} \mathrm{x} _{1}^{\mathrm{n} 1} \mathrm{x} _{2}^{\mathrm{n} 2} \ldots \ldots \mathrm{x} _{\mathrm{k}}^{\mathrm{nk}}$,

Where $\mathrm{n} _{1}+\mathrm{n} _{2}+\ldots \ldots+\mathrm{n} _{\mathrm{k}}=\mathrm{n}$ and $0 \leq \mathrm{n} _{1}, \mathrm{n} _{2}, \ldots \ldots . \mathrm{n} _{\mathrm{k}} \leq \mathrm{n}$

• The greatest coefficient in this expansion is $\frac{\mathrm{n} !}{(\mathrm{q} !)^{k-\mathrm{r}}((\mathrm{q}+1) !)^{\mathrm{r}}}$ where $\mathrm{q}$ is the quotient and $\mathrm{r}$ is the remainder when $\mathrm{n}$ is divided by $\mathrm{k}$.

$\quad \quad $ Eg. Find the greatest coefficient in $(x+y+z+w)^{15}$

$\quad \quad $ $\mathrm{n}=15, \mathrm{k}=4$ we have $15=4 \times 3+3$ i.e. $\mathrm{q}=3, \mathrm{r}=3$ greatest coefficient $=\frac{15 !}{(3 !)^{1}(4 !)^{3}}$

• Number of distinct terms in the expansion is ${ }^{n+k-1} C _{k-1}$ (Total number of terms).

• Number of positive integer solutions of $\mathrm{x} _{1}+\mathrm{x} _{2}+\ldots+\mathrm{x} _{\mathrm{k}}=\mathrm{n}$ is ${ }^{\mathrm{n}-1} \mathrm{C} _{\mathrm{k}-1}$.

• Number of non negative integer solutions of $x _{1}+x _{2}+\ldots+x _{k}=n$ is ${ }^{n+k-1} C _{k-1}$.

• Sum of all the coefficients is obtained by setting $\mathrm{x} _{1}=\mathrm{x} _{2}=\ldots \ldots \mathrm{x} _{\mathrm{k}}=1$.

Greatest Coefficient and Greatest term

Consider the binomial expansion of $(x+y)^{n}$. where $n \in W$. For a given value of $n$,

Maximum value of ${ }^{n} \mathrm{C} _{\mathrm{r}}$ is ${ }^{n} \mathrm{C} _{\mathrm{n} 2}$ if $\mathrm{n}$ is even

Maximum value of ${ }^{n} C _{r}$ is ${ }^{n} C _{\frac{n-1}{2}}={ }^{n} C _{\frac{n+1}{2}}$ if $n$ is od(d).

To find the greatest term in the expansion of $(x+y)^{n}$,

(i) Find $m=\frac{n+1}{1+\frac{x}{y}}$

(ii) If $\mathrm{m}$ is an integer we have $\mathrm{m}^{\text {th }}$ and $(\mathrm{m}+1)^{\text {th }}$ terms as greatest terms.

(iii) If $\mathrm{m}$ is not an integer, then $([\mathrm{m}]+1)$ th term is the greatest term where [.] denote the greatest integer $\leq \mathrm{m}$.

Divisibility Problems

From the expansion

$(1+\mathrm{a})^{\mathrm{n}}=1+{ }^{\mathrm{n}} \mathrm{C} _{1} \mathrm{a}+{ }^{\mathrm{n}} \mathrm{C} _{2} \mathrm{a}^{2}+\ldots . .+{ }^{\mathrm{n}} \mathrm{C} _{\mathrm{n}} \mathrm{a}^{\mathrm{n}}$, we can see that

(i) $\quad(1+a)^{n}-1$ is a mutliple of $a=M(a)$

(ii) $\quad (1+\mathrm{a})^{\mathrm{n}}-1-$ na is a mutliple of $\mathrm{a}^{2}=\mathrm{M}\left(\mathrm{a}^{2}\right)$

(iii) $\quad (1+\mathrm{a})^{\mathrm{n}}-1-\mathrm{na}-\frac{\mathrm{n}(\mathrm{n}-1)}{2} \mathrm{a}^{2}$ is a mutliple of $\mathrm{a}^{3}$ and so on.

For example

• $(1+8)^{50}-1=9^{\mathrm{n}}-1$ is $\mathrm{M}(8)$

• $(1+8)^{50}-1-50 \times 8=9^{\mathrm{n}}-399$ is $\mathrm{M}\left(8^{2}\right)=\mathrm{M}(64)$

• $(1+8)^{50}-1-50 \times 8-\frac{50 \times 49}{2} 8^{2}=\mathrm{M}\left(8^{3}\right)=\mathrm{M}(512)$ and soon

Binomial Theorem for any index (for negative or fractional index)

$\quad$ If $n \varepsilon Q$, then $(1+x)^{n}=1+n x+\frac{n(n-1)}{2 !} x^{2}+\frac{n(n-1)(n-2)}{3 !} x^{3}+\ldots . . \infty$ provided $|x|<1$.

• For any index $n$, the general term in the expansion of

i. $\quad(1+x)^{n}$ is $T _{r+1}=\frac{n(n-1) \ldots \ldots . .(n-r+1)}{r !} x^{r}$

ii. $\quad(1+\mathrm{x})^{-\mathrm{n}}$ is $\mathrm{T} _{\mathrm{r}+1}=\frac{(-1)^{\mathrm{r}} \mathrm{n}(\mathrm{n}+1) \ldots \ldots \ldots .(\mathrm{n}+\mathrm{r}-1)}{\mathrm{r} !} \mathrm{x}^{\mathrm{r}}$

iii. $\quad(1-\mathrm{x})^{\mathrm{n}}$ is $\mathrm{T} _{\mathrm{r}+1}=\frac{(-1)^{\mathrm{r}} \mathrm{n}(\mathrm{n}-1) \ldots \ldots \ldots .(\mathrm{n}-\mathrm{r}+1)}{\mathrm{r} !} \mathrm{x}^{\mathrm{r}}$

iv. $\quad(1-\mathrm{x})^{-\mathrm{n}}$ is $\mathrm{T} _{\mathrm{r}+1}=\frac{\mathrm{n}(\mathrm{n}+1) \ldots \ldots \ldots .(\mathrm{n}+\mathrm{r}-1)}{\mathrm{r} !} \mathrm{x}^{\mathrm{r}}$

• The following expansions should be remembered (for $|\mathrm{x}|<1$ ).

i. $\quad (1+x)^{-1}=1-x+x^{2}-x^{3}+\ldots \ldots \ldots \infty$

ii. $\quad (1-x)^{-1}=1+x+x^{2}+x^{3}+$ $\ldots\ldots \ldots\infty$

iii. $\quad (1+x)^{-2}=1-2 x+3 x^{2}-4 x^{3}+$ $\ldots\ldots \ldots\infty$

iv. $\quad (1-x)^{-2}=1+2 x+3 x^{2}+4 x^{3}+$ $\ldots\ldots \ldots\infty$

• Note : The expansion in ascending powers of $x$ is valid if $x$ is small. If $x$ is large (i.e. $|x|>1)$, then we may find it convenient to expand in powers of $\frac{1}{x}$, which then will be small.

Exponential series

• $ \mathrm{e}^{\mathrm{x}}=1+\frac{\mathrm{x}}{1 !}+\frac{\mathrm{x}^{2}}{2 !}+\frac{\mathrm{x}^{3}}{3 !}+\ldots . . \infty$

• $ \mathrm{e}=1+\frac{1}{1 !}+\frac{1}{2 !}+\frac{1}{3 !}+\ldots . \infty(\mathrm{e} \simeq 2.72)$

• $ \mathrm{e}+\mathrm{e}^{-1}=2\left(1+\frac{1}{2 !}+\frac{1}{4 !}+\frac{1}{6 !}+\ldots . \infty\right)$

• $ \mathrm{e}-\mathrm{e}^{-1}=2\left(\frac{1}{1 !}+\frac{1}{3 !}+\frac{1}{5 !}+\frac{1}{7 !}+\ldots . \infty\right)$

Logarithmic series

For $-1<\mathrm{x} \leq 1$

$\log _{e}(1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\ldots . . \infty$

• $ \log _{\mathrm{e}}(1-\mathrm{x})=-\mathrm{x}-\frac{\mathrm{x}^{2}}{2}-\frac{\mathrm{x}^{3}}{3}-\frac{\mathrm{x}^{4}}{4}+\ldots . \infty,-1 \leq \mathrm{x}<1$

• $ \log \left(\frac{1+\mathrm{x}}{1-\mathrm{x}}\right)=2\left(\mathrm{x}+\frac{\mathrm{x}^{3}}{3}+\frac{\mathrm{x}^{5}}{5}+\ldots . \infty\right),-1<\mathrm{x}<1$

• $\log _{\mathrm{e}} 2=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots . . \infty \approx 0.693$

Solved examples

1. If $\mathrm{C} _{0}, \mathrm{C} _{1}, \mathrm{C} _{2}, \ldots . . \mathrm{Cn}$ denote the coefficients in the binomial expansion of $(1+\mathrm{x})^{\mathrm{n}}$, prove that :

$ \mathrm{C} _{1}+2 \mathrm{C} _{2}+3 \mathrm{C} _{3}+\ldots . .+\mathrm{nC}=\mathrm{n} \cdot 2^{\mathrm{n}-1} $

i.e. $ \sum _{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r} \cdot \mathrm{C} _{\mathrm{r}}=\mathrm{n} \cdot 2^{\mathrm{n}-1}$

2. If $\mathrm{C} _{0}, \mathrm{C} _{1}, \mathrm{C} _{2}, \ldots . . . \mathrm{Cn}$ denote the coefficients in the binomial expansion of $(1+\mathrm{x})^{\mathrm{n}}$, prove that: $\mathrm{C} _{0}+3 \mathrm{C} _{1}+5 \mathrm{C} _{2}+\ldots . .+(2 \mathrm{n}+1) \mathrm{C} _{\mathrm{n}}=(\mathrm{n}+1) \cdot 2^{\mathrm{n}}$.

3. If $\mathrm{C} _{0}, \mathrm{C} _{1}, \mathrm{C} _{2}, \ldots . . \mathrm{Cn}$ denote the coefficients in the binomial expansion of $(1+\mathrm{x})^{\mathrm{n}}$, prove that: $1^{2} \cdot \mathrm{C} _{1}+2^{2} \cdot \mathrm{C} _{2}+3^{2} \cdot \mathrm{C} _{3}+\ldots .+\mathrm{n}^{2} \cdot \mathrm{C} _{\mathrm{n}}=\mathrm{n}(\mathrm{n}+1) 2^{\mathrm{n}-2}$

4. If $\mathrm{C} _{0}, \mathrm{C} _{1}, \mathrm{C} _{2}, \ldots . . \mathrm{Cn}$ denote the coefficients in the binomial expansion of $(1+\mathrm{x})^{\mathrm{n}}$, prove that: $1^{3} \cdot \mathrm{C} _{1}+2^{3} \cdot \mathrm{C} _{2}+3^{3} \cdot \mathrm{C} _{3}+\ldots . .+\mathrm{n}^{3} \cdot \mathrm{C} _{\mathrm{n}}=\mathrm{n}^{2}(\mathrm{n}+3) 2^{\mathrm{n}-3}$

5. If $\mathrm{C} _{0}, \mathrm{C} _{1}, \mathrm{C} _{2} \ldots \ldots, \mathrm{C} _{\mathrm{n}-1}, \mathrm{C} _{\mathrm{n}}$ denote the binomial coefficients in the expansion of $(1+\mathrm{x})^{\mathrm{n}}$, prove that: $\frac{\mathrm{C} _{1}}{\mathrm{C} _{0}}+2 \cdot \frac{\mathrm{C} _{2}}{\mathrm{C} _{1}}+3 \cdot \frac{\mathrm{C} _{3}}{\mathrm{C} _{2}}+\ldots \ldots+\mathrm{n} \cdot \frac{\mathrm{C} _{\mathrm{n}}}{\mathrm{C} _{\mathrm{n}-1}}=\frac{\mathrm{n}(\mathrm{n}+1)}{2}$

6. If $\mathrm{C} _{0}, \mathrm{C} _{1}, \mathrm{C} _{2} \ldots . ., \mathrm{C} _{\mathrm{n}-1}, \mathrm{C} _{\mathrm{n}}$ denote the binomial coefficients in the expansion of $(1+\mathrm{x})^{\mathrm{n}}$, prove that: $\left(\mathrm{C} _{0}+\mathrm{C} _{1}\right)\left(\mathrm{C} _{1}+\mathrm{C} _{2}\right)\left(\mathrm{C} _{2}+\mathrm{C} _{3}\right)\left(\mathrm{C} _{3}+\mathrm{C} _{4}\right) \ldots \ldots\left(\mathrm{C} _{\mathrm{n}-1}+\mathrm{C} _{\mathrm{n}}\right)=\frac{\mathrm{C} _{0} \mathrm{C} _{1} \mathrm{C} _{2} \ldots . \mathrm{C} _{\mathrm{n}-1}(\mathrm{n}+1)^{\mathrm{n}}}{\mathrm{n} !}$

7. If $\mathrm{C} _{0}, \mathrm{C} _{1}, \mathrm{C} _{2}, \ldots . . \mathrm{Cn}$ denote the binomial cofficients in the expansion of $(1+\mathrm{x})^{\mathrm{n}}$, prove that $\mathrm{C} _{0}+\frac{\mathrm{C} _{1}}{2}+\frac{\mathrm{C} _{2}}{3}+\ldots \ldots+\frac{\mathrm{C} _{\mathrm{n}}}{\mathrm{n}+1}=\frac{2^{\mathrm{n}+1}-1}{\mathrm{n}+1}$

Practice questions

1. The sum $\frac{1}{1 !(\mathrm{n}-1) !}+\frac{1}{3 !(\mathrm{n}-3) !}+\frac{1}{5 !(\mathrm{n}-5) !}+\ldots \ldots \ldots=$

2. If the coefficient of $x^{n}$ in $(1+x)^{101}\left(1-x+x^{2}\right)^{100}$ is non-zero, then $n$ cannot be of the form

(a). $3 \mathrm{r}+1$

(b). $3 \mathrm{r}$

(c). $3 \mathrm{r}+2$

(d). none of these

3. The coefficient of $x^{r} ; 0 \leq r \leq n-1$, is the expansion of $(x+3)^{n-1}+(x+3)^{n-2}(x+2)+(x+3)^{n-2}$ $(\mathrm{x}+2)^{2}+\ldots . .(\mathrm{x}+2)^{\mathrm{n}-1}$ are

(a). ${ }^{n} C _{r}\left(3^{r}-2^{n}\right)$

(b). ${ }^{n} C _{r}\left(3^{n-r}-2^{n-r}\right)$

(c). ${ }^{n} C _{r}\left(3^{r}-2^{n-r}\right)$

(d). none of these

4. The number of real negative terms in the binomial expension of $(1+i x)^{4 n-2}, n \in N, x>0$ is

(a). $\mathrm{n}$

(b). $n+1$

(c). $\mathrm{n}-1$

(d). $2 \mathrm{n}$

5. $(\mathrm{n}+2){ }^{n} C _{0} 2^{\mathrm{n}+1}-(\mathrm{n}+1){ }^{\mathrm{n}} \mathrm{C} _{1} 2^{\mathrm{n}}+\mathrm{n} \cdot{ }^{n} \mathrm{C} _{2} 2^{\mathrm{n}-1} \ldots . .$. is equal to

(a). $4$

(b). $4 \mathrm{n}$

(c). $4(\mathrm{n}+1)$

(d). $2(\mathrm{n}+2)$

6. $\sum _{\mathrm{k}=1}^{\infty} \mathrm{k}\left(1+\frac{1}{\mathrm{n}}\right)^{\mathrm{k}-1}=$

(a). $\mathrm{n}(\mathrm{n}-1)$

(b). $\mathrm{n}(\mathrm{n}+1)$

(c). $n^{2}$

(d). $(\mathrm{n}+1)^{2}$

7. The sum of rational term in $(\sqrt{2}+\sqrt[3]{3}+\sqrt[6]{5})^{10}$ is equal to

(a). 12632

(b). 1260

(c). 126

(d). none of these

8. Last two digit of $(23)^{14}$ are

(a). 01

(b). 03

(c). 09

(d). none of these

9. If $(4+\sqrt{15})^{\mathrm{n}}=\mathrm{I}+\mathrm{f}$, where $\mathrm{n}$ is an odd natural number, $\mathrm{I}$ is an interger and $0<\mathrm{f}<1$, then

(a). I is a natural number

(b). I an even integer

(c). $(\mathrm{I}+\mathrm{f})(1-\mathrm{F})=1$

(d). $ 1-\mathrm{f}=(4+\sqrt{5})^{\mathrm{n}}$

10. The number of rational numbers lying in the interval $(2002, 2003)$ all whose digits after the decimal point are non-zero and are in decreasing order is

(a). $ \sum _{\mathrm{i}=1}^{9}{ }^{9} \mathrm{P} _{\mathrm{i}}$

(b). $ \sum _{\mathrm{i}=1}^{10}{ }^{9} \mathrm{P} _{\mathrm{i}}$

(c). $ 2^{9}-1$

(d). $2^{10}-1$

11. Match the following: