Statement of binomial theorem :

PYQ-2023-Binomial_Theorem-Q4, PYQ-2023-Binomial_Theorem-Q8, PYQ-2023-Binomial_Theorem-Q11, PYQ-2023-Binomial_Theorem-Q14, PYQ-2023-Binomial_Theorem-Q15

$\quad$ If $a, b \in R$ and $n \in N$, then,

$$(a+b)^{n}={ }^{n} C_{0} a^{n} b^{0}+{ }^{n} C_{1} a^{n-1} b^{1}+{ }^{n} C_{2} a^{n-2} b^{2}+\ldots+{ }^{n} C_{r} a^{n-r} b^{r}+\ldots+{ }^{n} C_{n} a^{0} b^{n}=\sum_{r=0}^{n}{ }^{n} C_{r} a^{n-r} b^{r}$$

Properties of binomial theorem:

PYQ-2023-Binomial_Theorem-Q10, PYQ-2023-Binomial_Theorem-Q11

• General term : $T_{r+1}={ }^{n} C_{r} a^{n-r} b^{r}$

• Middle term (s) :

• If $n$ is even, there is only one middle term, which is $\left(\frac{n+2}{2}\right)$ th term.

• If $\mathrm{n}$ is odd, there are two middle terms, which are $\left(\frac{\mathrm{n}+1}{2}\right)$ th and $\left(\frac{\mathrm{n}+1}{2}+1\right)$ th terms.

Multinomial theorem :

$$\left(x_{1}+x_{2}+x_{3}+\ldots \ldots \ldots . x_{k}\right)^{n}=\sum_{r_{1}+r_{2}+\ldots+r_{k}=n} \frac{n !}{r_{1} ! r_{2} ! \ldots r_{k} !} x_{1}^{r_{1}} \cdot x_{2}^{r_{2}} \ldots x_{k}^{r_{k}}$$

Here, total number of terms in the expansion $={ }^{n+k-1} C_{k-1}$

Application of binomial theorem :

• If $(\sqrt{A}+B)^{n}=I+f$ where $I$ and $n$ are positive integers, $n$ being odd and $0<f<1$ then,

$$(I+f) f=k^{n}, \text{where}, A-B^{2}=k>0 \text{and} \sqrt{A}-B<1$$

• If $\mathbf{n}$ is an even integer, then $(I+f)(1-f)=k^{n}$

Properties of binomial coefficients :

PYQ-2023-Binomial_Theorem-Q1, PYQ-2023-Binomial_Theorem-Q7

• $${ }^{n} C_{0}+{ }^{n} C_{1}+{ }^{n} C_{2}+\ldots \ldots . .+{ }^{n} C_{n}=2^{n}$$

• $${ }^{n} C_{0}-{ }^{n} C_{1}+{ }^{n} C_{2}-{ }^{n} C_{3}+\ldots \ldots \ldots \ldots .+(-1)^{n}{ }^{n} C_{n}=0$$

• $${ }^{n} C_{0}+{ }^{n} C_{2}+{ }^{n} C_{4}+\ldots .={ }^{n} C_{1}+{ }^{n} C_{3}+{ }^{n} C_{5}+\ldots .=2^{n-1}$$

• $${ }^{n} C_{r}+{ }^{n} C_{r-1}={ }^{n+1} C_{r} $$

• $$\frac{{ }^{n} C_{r}}{{ }^{n} C_{r-1}}=\frac{n-r+1}{r}$$

Binomial theorem for negative integer or fractional indices:

$$(1+x)^{n}=1+n x+\frac{n(n-1)}{2 !} x^{2}+\frac{n(n-1)(n-2)}{3 !}x^{3}+\ldots .+\frac{n(n-1)(n-2) \ldots \ldots .(n-r+1)}{r !} x^{r}+\ldots .,|x|<1 .$$

Binomial theorem for positive integer:

$\quad$ If $n$ is any positive integer, then,

$$ \begin{aligned} & (x+a)^{n}={ }^{n} C_{0} x^{n}+{ }^{n} C_{1} x^{n-1} a+{ }^{n} C_{2} x^{n-2} a^{2}+\ldots+{ }^{n} C_{n} a^{n} . \ & (x+a)^{n}=\sum_{r=0}^{n}{ }^{n} C_{r} x^{n-r} a^{r} \qquad \qquad\text{(~This is called binomial theorem)}\end{aligned} $$

$\quad$ Here, $^nC_0, ^nC_1, ^nC_2, \ldots, ^nC_n$ are called binomial coefficients and $^nC_r = \frac{n!}{r!(n-r)!}$ for $0 \leq r \leq n$.

Properties of binomial theorem for positive integer:

PYQ-2023-Binomial_Theorem-Q5, PYQ-2023-Binomial_Theorem-Q13, PYQ-2023-Binomial_Theorem-Q16, PYQ-2023-Binomial_Theorem-Q18, PYQ-2023-Binomial_Theorem-Q19, PYQ-2023-Binomial_Theorem-Q22, PYQ-2023-Binomial_Theorem-Q23

• Total number of terms in the expansion of $(x+a)^{n}$ is $(n+1)$.

• The sum of the indices of $x$ and $a$ in each term is $n$.

• The above expansion is also true when $\mathrm{x}$ and a are complex numbers.

• The coefficient of terms equidistant from the beginning and the end are equal.

$\quad \quad$ These coefficients are known as the binomial coefficients and ${ }^{n} C_{r}={ }^{n} C_{n-r}, r=0,1,2, \ldots, n$.

• General term in the expansion of $(x+c)^{n}$ is given by $T_{r+1}={ }^{n} C_{r} x^{n-r} a^{r}$.

• The values of the binomial coefficients steadily increase to maximum and then steadily decrease .

$$ \begin{aligned} & \begin{array}{l} (x-a)^{n}={ }^{n} C_{0}-{ }^{n} C_{1} x^{n-1} a+{ }^{n} C_{2} x^{n-2} a^{2}-{ }^{n} C_{3} x^{n-3} a^{3}+\ldots +(-1)^{n}{ }^{n} C_{n} a^{n} \end{array} \\ & \text { i.e., }(x-a)^{n}=\sum_{r=0}^{n}(-1)^{r}{ }^{n} C_{r} \cdot x^{n-r} \cdot a^{r} \\ & \qquad(1+x)^{n}={ }^{n} C_{0}+{ }^{n} C_{1} x+{ }^{n} C_{2} x^{2}+\ldots+{ }^{n} C_{n} x^{n} \\ & \text { i.e., } \quad(1+x)^{n}=\sum_{r=0}^{n}{ }^{n} C_{r} \cdot x^{r} \end{aligned} $$

• The coefficient of $x^{r}$ in the expansion of $(1+x)^{n}$ is ${ }^{n} C_{r}$.

• $(x+a)^{n}+(x-a)^{n}=2\left({ }^{n} C_{0} x^{n} a^{0}+{ }^{n} C_{2} x^{n-2} a^{2}+\ldots\right)$ and $(x+a)^{n}-(x-a)^{n}=2\left({ }^{n} C_{1} x^{n-1} a+{ }^{n} C_{3} x^{n-3} a^{3}+\ldots\right)$

  • If $\mathrm{n}$ is odd, then $(\mathrm{x}+\mathrm{a})^{\mathrm{n}}+(\mathrm{x}-\mathrm{a})^{\mathrm{n}}$ and $(\mathrm{x}+\mathrm{a})^{\mathrm{n}}-(\mathrm{x}-\mathrm{a})^{\mathrm{n}}$ both have the same number of terms equal to $(n+1 / 2)$.

  • If $n$ is even, then $(x+a)^{n}+(x-a)^{n}$ has $(n+1 / 2)$ terms. and $(x+a)^{n}-(x-a)^{n}$ has $(n / 2)$ terms.

• In the binomial expansion of $(x+a)^{n}$, the $r$ th term from the end is $(n-r+2)$ th term

$$ \begin{array}{r} \begin{array}{r} (1-x)^{n}={ }^{n} C_{0}-{ }^{n} C_{1} x+{ }^{n} C_{2} x^{2}-{ }^{n} C_{3} x^{3}+\ldots+(-1)^{r} C_{r} x^{r} \ +\ldots+(-1)^{n}{ }^{n} C_{n} x^{n} \end{array} \ \text { i.e., } \quad(1-x)^{n}=\sum_{r=0}^{n}(-1)^{r}{ }^{n} C_{r} \cdot x^{r} \end{array} $$

• If $n$ is a positive integer, then number of terms in $(x+y+z)^{n}$ is $(n+l)(n+2) / 2$.

Middle term in the expansion of $(1+x)^{n}$:

PYQ-2023-Binomial_Theorem-Q11

• It $n$ is even, then in the expansion of $(x+a)^{n}$, the middle term is $(\frac{n}{2} +1)^{\text {th }}$ terms.

• If $n$ is odd, then in the expansion of $(x+a)^{n}$, the middle terms are $\frac{(n+1)}{2}$ th term and $\frac{(n+3)}{2}$ th term.

Greatest coefficient:

• If $n$ is even, then in $(x+a)^{n}$, the greatest coefficient is ${ }^{n} C_{\frac{n}{2}}.$

• If n is odd, then in $(x+a)^{n}$, the greatest coefficient is ${ }^{n} C_{\frac{n-1}{2}}$ or ${ }^{n} C_{\frac{n+1}{2}}$ both being equal.

10. Greatest term in the expansion of $(x+a)^{n}:$

• If $\frac{n+1}{1+\frac{x}{a}}$ is an integer $=p$ (say), then greatest term is $T_{p}==T_{p+1}$.

• If $\frac{n+1}{1+\frac{x}{a}}$ is not an integer with $m$ as integral part of $\frac{n+1}{1+\frac{x}{a}}$, then $T_{m+1}$. is the greatest term.

Important results on binomial coefficients :

PYQ-2023-Binomial_Theorem-Q2, PYQ-2023-Binomial_Theorem-Q3, PYQ-2023-Binomial_Theorem-Q20

• ${ }^{n} C_{r}+{ }^{n} C_{r-1}={ }^{n+1} C_{r}$

• $\frac{{ }^{n} C_{r}}{{ }^{n-1} C_{r-1}}=\frac{n}{r}$

• $\frac{{ }^{n} C_{r}}{{ }^{n} C_{r-1}}=\frac{n-r+1}{r}$

• $C_{0}+C_{1}+C_{2}+\ldots+C_{n}=2^{n}$

• $C_{0}+C_{2}+C_{4}+\ldots=C_{1}+C_{3}+C_{3}+\ldots=2^{n-1}$

• $C_{0}-C_{1}+C_{2}-C_{3}+\ldots+(-1)^{n} C_{n}=0$

• $C_{0} C_{r}+C_{1} C_{r+1}+\ldots+C_{n-r} C_{n}={ }^{2 n} C_{n+r}=\frac{(2 n) !}{(n-r) !(n+r) !}$

• $C_{0}^{2}+C_{1}^{2}+C_{2}^{2}+\ldots+C_{n}^{2}={ }^{2 n} C_{n}=\frac{(2 n) !}{(n !)^{2}}$

• $C_{0}-C_{2}+C_{4}-C_{6}+\ldots=(\sqrt{2})^{n} \cos \frac{n \pi}{4} $

• $C_{1}-C_{3}+C_{5}-C_{7}+\ldots=(\sqrt{2})^{n} \sin \frac{n \pi}{4} $

• $C_{0}-C_{1}+C_{2}-C_{3}+\ldots+(-1)^{r} C_{r}=(-1)^{r} n-1 C_{r}, r<n $

• $C_0^2 - C_1^2 + C_2^2 - C_3^2 + \ldots = \begin{cases} 0, & \text{if } n \text{ is odd.} \\ (-1)^{n/2} \cdot ^nC_{n/2}, & \text{if } n \text{ is even.} \end{cases}$

• $C_{0}+\frac{C_{1}}{2}+\frac{C_{2}}{3}+\ldots+\frac{C_{n}}{n+1}=\frac{2^{n+1}-1}{(n+1)} $

• $C_{0}-\frac{C_{1}}{2}+\frac{C_{2}}{3}-\frac{C_{3}}{4}+\ldots+(-1)^{n} \frac{C_{n}}{n+1}=\frac{1}{n+1} $

• $C_{0}+\frac{C_{1}}{2}+\frac{C_{2}}{2^{2}}+\frac{C_{3}}{2^{3}}+\ldots+\frac{C_{n}}{2^{n}}=\left(\frac{3}{2}\right)^{n} $

• $\sum_{r=0}^{n}(-1)^{r} C_{r}{\frac{1}{2^{r}}+\frac{3^{r}}{2^{2 r}}+\frac{7^{r}}{2^{3 r}}+\frac{15^{r}}{2^{4 r}}+\ldots \text{ upto } m \text{ terms }}=\frac{2^{mn}-1}{2^{mn}(2^n-1)} $

Divisibility problems:

PYQ-2023-Binomial_Theorem-Q17, PYQ-2023-Binomial_Theorem-Q21

$\quad$ From the expansion, $(1+x)^{n}=1+{ }^{n} C_{1} x+{ }^{n} C_{1} x^{2}+\ldots+{ }^{n} C_{n} x^{n}$

$\quad$ We can conclude that,

• $(1+x)^n-1 = ^nC_1 x + ^nC_2 x^2 + \ldots + ^nC_n x^n$ is divisible by $x$, i.e., it is a multiple of $x$.

$$(1+\mathrm{x})^{\mathrm{n}}-1=\mathrm{M}(\mathrm{x})$$

$$ (1+x)^{n}-1-n x={ }^{n} C_{2} x^{2}+{ }^{n} C_{3} x^{3}+\ldots+{ }^{n} C_{n} x^{n}=M\left(x^{2}\right) $$

$$ \begin{aligned} (1+x)^{n}-1-n x-\frac{n(n-1)}{2} x^{2} & ={ }^{n} C_{3} x^{3}+{ }^{n} C_{4} x^{4}+\ldots+{ }^{n} C_{n} x^{n} \ & =M\left(x^{3}\right) \end{aligned} $$

Multinomial theorem:

PYQ-2023-Binomial_Theorem-Q6

$\quad$ For any $\mathrm{n} \in \mathrm{N}$,

$$ \left(x_{1}+x_{2}\right)^{n}=\sum_{r_{1}+r_{2}=n} \frac{n !}{r_{1} ! r_{2} !} x_{1}^{r_{1}} x_{2}^{r_{2}} $$

$$ \left(x_{1}+x_{2}+\ldots+x_{n}\right)^{n}=\sum_{r_{1}+r_{2}+\ldots+r_{k}=n} \frac{n !}{r_{1} ! r_{2} ! \ldots r_{k} !} x_{1}^{r_{1}} x_{2}^{r_{2}} \ldots x_{k}^{r_{k}} $$

• The general term in the above expansion is :

$$ \frac{n !}{r_{1} ! r_{2} ! \ldots r_{k} !} x_{1}^{r_{1}} x_{2}^{r_{2}} \ldots x_{k}^{r_{k}} $$

• The greatest coefficient in The expansion of $(x_1 + x_2 + \ldots + x_m)^n$ is

$$\frac{n!}{(q!)^{m-r}[(q+1)!]^{r}}$$

$\quad$ when $n$ is divided by $m$, where $q$ and $r$ are the quotient and remainder, respectively.

• Number of Non-negative Integral Solutions

$\quad \quad \quad$ The number of non-negative integral solutions of the equation $(x_{1}+x_{2}+\ldots+x_{n}=n)$ is given by ${ }^{n+r-1} C_{r-1}$.

Binomial theorem for any index:

$\quad$ If $n$ is any rational number, then the binomial theorem for any index can be expressed as:

$$ (1+x)^{n} = 1 + nx + \frac{n(n-1)}{1 \cdot 2} x^{2} + \frac{n(n-1)(n-2)}{1 \cdot 2 \cdot 3} x^{3} + \ldots, \text{ where } |x|<1 $$

• If in the above expansion, $n$ is any positive integer, then the series in RHS is finite otherwise infinite.

• General term in the expansion of $(1+x)^{n}$ is :

$$ T_{r+1}=\frac{n(n-1)(n-2) \ldots[n-(r-1)]}{r!} x{ }^{r} $$

• Expansion of $(x+a)^{n}$ for any rational index:

Case I. When $x > a$, i.e., if $\frac{a}{x} < 1$, then:

$$(x + a)^n = x\left(1 + \frac{a}{x}\right)^n = x^n\left(1 + \frac{a}{x}\right)^n $$

$$ x^n\left(1 + \frac{a}{x}\right)^n= x^n[1 + n \cdot \frac{a}{x} + \frac{n(n-1)}{2!}(\frac{a}{x})^2 + \frac{n(n-1)(n-2)}{3!}(\frac{a}{x})^3 + \ldots ]$$

Case II. When $x < a$, i.e., $\frac{x}{a} < 1$:

$$(x + a)^n = a\left(1 + \frac{x}{a}\right)^n = a^n\left(1 + \frac{x}{a}\right)^n$$

$$ a^n(1 + \frac{x}{a})^n= a^n[1 + n \cdot \frac{x}{a} + \frac{n(n-1)}{2!}(\frac{x}{a})^2 + \frac{n(n-1)(n-2)}{3!}(\frac{x}{a})^3 + \ldots] $$

• $$(1-x)^{-n}=1+n x+\frac{n(n+1)}{1 \cdot 2} x^{2}+\frac{n(n+1)(n+2)}{1 \cdot 2 \cdot 3} x^{3}+\ldots=1+{ }^{n} C_{1} x+{ }^{(n+1)} C_{2} x^{2}+{ }^{(n+2)} C_{3} x^{3}+\ldots$$

$$(1+x)^{-n}=1-n x+\frac{n(n+1)}{2 !} x^{2}-\frac{n(n+1)(n+2)}{3 !} x^{3}+\ldots+(-1)^{r} \frac{n(n+1)(n+2) \ldots(n+r-1)}{r !} x^{r}+\ldots $$

• $$(1-x)^{n}=1-n x+\frac{n(n-1)}{2 !} x^{2}-\frac{n(n-1)(n-2)}{3 !} x^{3}+\ldots+(-1)^{r} \frac{n(n-1)(n-2) \ldots(n-r+1)}{r !} x^{r}+. . $$

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

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

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

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

• $$(1+x)^{-3}=1-3 x+6 x^{2}-\ldots \infty$$

• $$(1-x)^{-3}=1+3 x+6 x^{2}-\ldots \infty$$

• $(1+\mathrm{x})^{\mathrm{n}}=1+\mathrm{nx}$, if $\mathrm{x}^{2}, \mathrm{x}^{3}, \ldots$ are all very small as compared to $\mathrm{x}$.

Important results:

PYQ-2023-Binomial_Theorem-Q4, PYQ-2023-Binomial_Theorem-Q9, PYQ-2023-Binomial_Theorem-Q12, PYQ-2023-Binomial_Theorem-Q18

• Coefficient of $x^{m}$ in the expansion of $\left(a x^{p}+\frac{b}{x^{q}}\right)^{n}$ is the coefficient of $T_{r+1}$ where $r = np - \frac{m}{p + q}$

• The term independent of $x$ in the expansion of $(a x^{p}+\frac{b}{x^{q}})^{n}$ is the coefficient of $T_{r+1}$ where $r = \frac{np}{p + q}$

• If the coefficient of $r$ th, $(r+1)$ th and $(r+2)$ th term of $(1+x)^{n}$ are in AP, then $n^{2}-(4 r+1) n$ $+4 r^{2}=2$

• In the expansion of $(x+a)^{n}$

$$\frac{T_{r+1}}{T_{r}} = \frac{n - r + 1}{r} \cdot \frac{a}{x}$$

$\quad$ $\quad$ • The coefficient of $x^{n-1}$ in the expansion of

$$(x-1)(x-2) \ldots(x-n)=-\frac{n(n+1)}{2}$$

$\quad$ $\quad$ • The coefficient of $x^{n-1}$ in the expansion of

$$(x+1)(x+2) \ldots(x+n)=\frac{n(n+1)}{2}$$

• If the coefficient of pth and qth terms in the expansion of $(1+x)^{n}$ are equal, then $p+q=n$ $+2$

• If the coefficients of $x^{r}$ and $x^{r+1}$ in the expansion of $\left(a + \frac{x}{b}\right)^{n} $ are equal, then, $\mathrm{n}=(\mathrm{r}+1)(\mathrm{ab}+1)-1$

• If $n$ is a positive integer and $a_{1}, a_{2}, \ldots, a_{m} \in C$, then the coefficient of $x^{r}$ in the expansion of $\left(a_{1}+a_{2} x+a_{3} x^{2}+\ldots+a_{m} x^{m-1}\right)^{n}$ is

$$ \sum \frac{n !}{n_{1} ! n_{2} ! \ldots n_{m} !} a_{1}^{n_{1}} x a_{2}^{n_{2}} \ldots a_{m}^{n_{m}} $$

• For $|x|<1$,

$\quad$ $\quad$ • $1+x+x^{2}+x^{3}+\ldots+\infty=1 / 1-x$

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

• Total number of terms in the expansion of $(a+b+c+d)^{n}$ is $\frac{(n+1)(n+2)(n+3)}{6}$.

Important points to be remembered :

• If $n$ is a positive integer, then $(1+x)^{n}$ contains $(n+1)$ terms i.e., a finite number of terms.

$\quad$ When $n$ is general exponent, then the expansion of $(1+x)^{n}$ contains infinitely many terms.

• When $n$ is a positive integer, the expansion of $(l+x)^{n}$ is valid for all values of $x$.

$\quad$ If $n$ is general exponent, the expansion of $(i+x)^{n}$ is valid for the values of $x$ satisfying the condition $|\mathrm{x}|<1$.

$\sum_{\mathrm{r}=0}^{\mathrm{n}} \mathrm{r}^2{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}=\mathrm{n}(\mathrm{n}+1) \cdot 2^{\mathrm{n}-2}$