Binomial Theorem Questions - JEE

Find the value of $({}^{10}C_0) + ({}^{10}C_0 + {}^{10}C_1) + ({}^{10}C_0 + {}^{10}C_1 + {}^{10}C_2) + \ldots + ({}^{10}C_0 + {}^{10}C_1 + {}^{10}C_2 + \ldots + {}^{10}C_9)$.

For Problems 15–17: Let $P = \displaystyle\sum_{r=1}^{50} \frac{{}^{50+r}C_r(2r-1)}{{}^{50}C_r(50+r)}$, $Q = \displaystyle\sum_{r=0}^{50} \left({}^{50}C_r\right)^2$, $R = \displaystyle\sum_{r=0}^{100} (-1)^r \left({}^{100}C_r\right)^2$15. The value of $P - Q$ is equal to

Let $m$ and $n$ be the coefficients of seventh and thirteenth terms respectively in the expansion of $\left(\frac{1}{3}x^{1/3} + \frac{1}{2x^{2/3}}\right)^{18}$. Then $\left(\frac{n}{m}\right)^{1/3}$ is:

For which of the following values of x, 5th term is the numerically greatest term in the expansion of $(1 + x/3)^{10}$:

We have $\dfrac{2^{403}}{15}$. Find the fractional part and the value of $k$ where the fractional part equals $\dfrac{k}{15}$.

The value of ${}^nC_1 + {}^{n+1}C_2 + {}^{n+2}C_3 + \ldots + {}^{n+m-1}C_m$ is equal to

If $p + q = 1$, then $\sum_{r=0}^{n} r^3 {}^nC_r p^r q^{n-r} =$

The coefficient of $x^n$ in the expansion of $(1+x)(1-x)^n$ is:

$\displaystyle\sum_{k=1}^{\infty} k\left(1-\dfrac{1}{n}\right)^{k-1} =$

Find $n$, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of $\left(\sqrt[4]{2} + \dfrac{1}{\sqrt[4]{3}}\right)^n$ is $\sqrt{6} : 1$.

If the third term in the binomial expansion of $(1+x^{\log_2 x})^5$ equals 2560, then a possible value of $x$ is:

The middle term in the expansion of $\left(\frac{x}{2} + \frac{1}{2x}\right)^{2n}$ is equal to

If number of terms in the expansion of $(1 + 2x - 3y + 4z)^n$ is 286, then find the coefficient of term containing $xyz$.

For natural numbers m, n if $(1 - y)^m (1 + y)^n = 1 + a_1 y + a_2 y^2 + \ldots$, and $a_1 = a_2 = 10$, then $(m, n)$ is

If the coefficient of $x^3$ in the expansion of $(1 + ax + bx^2)(1-2x)^{18}$ is zero, then which of the following pairs $(a, b)$ satisfies this condition?

If the term independent of x in the expansion \left(\frac{3}{2}x^2 - \frac{1}{3x}\right)^9 is k, then 18k is equal to

Sum the series: \[1 + \frac{3}{2^1} + \frac{1\cdot3}{1\cdot2}\cdot\frac{3^2}{2^6} + \frac{1\cdot3\cdot5}{1\cdot2\cdot3}\cdot\frac{3^3}{2^7} + \cdots \text{ to } \infty.\]

If some three consecutive coefficients in the binomial expansion of $(x + 1)^n$ in powers of $x$ are in the ratio $2 : 15 : 70$, then the average of these three coefficients is ___________.

The series {}^nC_1 + 1 \times {}^nC_2 + 2 \times {}^nC_3 + \cdots + n \times {}^nC_n is equal to

If \(|x|

The positive value of $\lambda$ for which the co-efficient of $x^2$ in the expression $x^2\left(\sqrt{x}+\dfrac{\lambda}{x^2}\right)^{10}$ is 720, is:

Value of $\displaystyle\sum_{k=1}^{\infty}\sum_{r=0}^{k} \dfrac{1}{3^k}\binom{k}{r}$ is

The expression \[\left(\sqrt{2x^2+1}+\sqrt{2x^2-1}\right)^6 + \left(\frac{2}{\sqrt{2x^2+1}+\sqrt{2x^2-1}}\right)^6\] is a polynomial of degree

The term independent of $x$ in the expansion of $\left(\dfrac{1}{60} - \dfrac{x^8}{81}\right)\cdot\left(2x^2 - \dfrac{3}{x^2}\right)^6$ is equal to ___________.

If $(1+x)^n = C_0 + C_1x + C_2x^2 + \cdots + C_nx^n$, then $C_0C_2 + C_1C_3 + C_2C_4 + \cdots + C_{n-2}C_n =$

Find the number of irrational terms in the expansion of $(\sqrt[8]{5} + \sqrt[6]{2})^{100}$.

Find the coefficient of $x^3$ in the expansion of $(1 + x + 2x^2)\left(2x^2 - \dfrac{1}{3x}\right)^9$.

Given $(1+x^{\log_2 x})^5$, its third term is $T_3 = 2560$. Find the value(s) of $x$.

Let $R = (5\sqrt{5} + 11)^{2n+1}$ and $f = R - [R]$ where $[\,]$ denotes the greatest integer function. Prove that $Rf = 4^{2n+1}$.

Given below are two statements: **Statement I:** $25^{13}+20^{13}+8^{13}+3^{13}$ is divisible by 7. **Statement II:** The integral part of $(7+4\sqrt{3})^{25}$ is an odd number. In the light of the above statements, choose the correct answer:

The value of $\displaystyle\sum_{r=2}^{10} {}^rC_2 \cdot {}^{10}C_r$ is

If in the expansion of $(a - 2b)^n$, the sum of 5th and 6th terms is 0, then the values of $a/b =$

$\left[({}^nC_0 + {}^nC_3 + \cdots) - (1/2)({}^nC_1 + {}^nC_2 + {}^nC_4 + {}^nC_5 + \cdots)\right]^2 + (3/4)({}^nC_1 - {}^nC_2 + {}^nC_4 - {}^nC_5 + \cdots)^2 =$

If $n$ is a positive integer, then $(\sqrt{3}+1)^{2n} - (\sqrt{3}-1)^{2n}$ is

$\displaystyle\sum_{r=0}^{300} a_r x^r = (1+x+x^2+x^3)^{100}$. If $a = \displaystyle\sum_{r=0}^{300} a_r$, then $\displaystyle\sum_{r=0}^{300} r\, a_r$ is equal to

The middle term in the expansion of $\left(\frac{x}{2} + \frac{2}{x}\right)^n$ is

If \binom{n}{4}, \binom{n}{5} and \binom{n}{6} are in AP, find the value of n.

The coefficient of $x^4$ in the expansion of $\left(\sqrt{1+x^2} - x\right)^{-1}$ in ascending powers of $x$, when \(|x|

If $a_n = \sum_{r=0}^{n} \frac{1}{\binom{n}{r}}$, then $\sum_{r=0}^{n} \frac{r}{\binom{n}{r}}$ is equal to

Number of ways = ${}^{21}C_0 + {}^{21}C_1 + {}^{21}C_2 + \cdots + {}^{21}C_{10}$Find the value of this sum.

Consider the following statements:Statement-1: $\sum_{r=0}^{n}(r+1)\,{}^nC_r = (n+2)\,2^{n-1}$Statement-2: $\sum_{r=0}^{n}(r+1)\,{}^nC_r\,x^r = (1+x)^n + nx(1+x)^{n-1}$Which of the following is true?

If the coefficients of $x^2$ and $x^3$ are both zero, in the expansion of the expression $(1 + ax + bx^2)(1 - 3x)^{15}$ in powers of $x$, then the ordered pair $(a, b)$ is equal to:

If the coefficients of pth, (p+1)th and (p+2)th terms in the expansion of $(1 + x)^n$ are in AP, then

If $\left(\dfrac{8}{x^2} + \dfrac{6}{x} + 4\right)^{10} = \sum_{r=0}^{20} a_r \left(\dfrac{2}{x}\right)^r$, then find the value of $\dfrac{a_7}{a_{13}}$.

The coefficients of $x^p$ and $x^q$ ($p, q \in \mathbb{N}$) in the expansion of $(1 + x)^{p+q}$ are

If $x$ is positive, the first negative term in the expansion of $(1+x)^{27/5}$ is (\(|x|

For Problems 15–17: Let $P = \displaystyle\sum_{r=1}^{50} \frac{{}^{50+r}C_r(2r-1)}{{}^{50}C_r(50+r)}$, $Q = \displaystyle\sum_{r=0}^{50} \left({}^{50}C_r\right)^2$, $R = \displaystyle\sum_{r=0}^{100} (-1)^r \left({}^{100}C_r\right)^2$17. The value of $Q + R$ is equal to

The sum $\displaystyle\sum_{0\le i\le j\le 10} \binom{10}{C_j}\binom{j}{C_i}$ is equal to

Let $(x + 10)^{50} + (x - 10)^{50} = a_0 + a_1x + a_2x^2 + \cdots + a_{50}x^{50}$, for all $x \in \mathbb{R}$; then $\dfrac{a_2}{a_0}$ is equal to ___________.

The fractional part of $2^{4n}/15$ is ($n \in \mathbb{N}$)

Binomial Theorem Questions (605)