Binomial Theorem Questions - JEE

Consider the binomial expansion R = (1 + 2x)n = I + f, where I is the integral part of R and f is the fractional part of R, n ∈ ℕ. Also, the sum of coefficients of R is 2187. The value of (n + Rf) for x = 1/2 is

Suppose $\displaystyle\sum_{r=0}^{2023}r^2\cdot{}^{2023}C_r=2023\times\alpha\times2^{2022}$. Then the value of $\alpha$ is ___.

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 coefficient of x^3y^4z^2 in the expansion of (2x - 3y + 4z)^9 is

The coefficient of a^3b^4c^5 in the expansion of (bc + ca + ab)^6 is

If $a$, $b$ and $c$ are the greatest values of $\binom{19}{p}$, $\binom{20}{q}$ and $\binom{21}{r}$ respectively, then

If $(1 + x - 2x^2)^6 = 1 + a_1 x + a_2 x^2 + \cdots + a_{12} x^{12}$, then the expression $a_2 + a_4 + a_6 + \cdots + a_{12}$ has the value

Find the coefficient of $x^{20}$ in the expansion of $(1 + x^2)^{40} \times \left(x^{-5/2} + 2 + \frac{2}{x^{1/2}}\right)$.

If the 4th term of (px + $\frac{1}{x}$)n is $\frac{5}{2}$, then np equals

The term independent of x in the expansion of $\left(\sqrt{\frac{x}{3}}+\frac{3}{2 x^{2}}\right)^{10}$ will be

The sum, of the coefficients of the first 50 terms in the binomial expansion of (1 - x )100, is equal to

If n ∈ ℕ, $\sum_{k=0}^{2n} (-1)^k ({}^{2n}C_k)^2 = A$, then find the value of $\sum_{k=0}^{2n} (-1)^k (k - 2n)({}^{2n}C_k)^2$.

The coefficient of $x^{18}$ in the product $(1 + x)(1 - x)^{10}(1 + x + x^2)^9$ is ___________.

The coefficient of $t^4$ in the expansion of $\left(\dfrac{1 - t^6}{1 - t}\right)^3$ is ___________.

Let $S_n = \binom{n}{0}\binom{n}{1} + \binom{n}{1}\binom{n}{2} + \ldots + \binom{n}{n-1}\binom{n}{n}$.If $\frac{S_{n+1}}{S_n} = \frac{15}{4}$, find the sum of all possible values of n (where $n \in \mathbb{N}$).

Find the term independent of $x$ in the expansion of $\left(\sqrt{\dfrac{x}{3}} + \dfrac{\sqrt{3}}{2x^2}\right)^{10}$.

If for some $m,n$; ${}^6C_m + 2({}^6C_{m+1}) + {}^6C_{m+2} > {}^8C_3$ and ${}^{n-1}P_3 : {}^nP_4 = 1:8$, then ${}^nP_{m+1} + {}^{n+1}C_m$ is equal to

The ratio of the coefficients of $x^{15}$ to the term independent of x in the expansion of $\left(x^2 + \frac{2}{x}\right)^{15}$ is:

In the binomial expansion of $(7^{1/5}-3^{1/10})^{60}$, find the number of irrational terms.

If $\binom{18}{r-2} + 2\binom{18}{r-1} + \binom{18}{r} \geq 13$, find the number of values of $r$.

For Problems 18–20: If $(1 + x + x^2)^{20} = a_0 + a_1 x + a_2 x^2 + \cdots + a_{40} x^{40}$, then answer the following questions.18. The value of $a_0 + a_1 + a_2 + \cdots + a_{19}$ is

If (1 + x + x^2 + x^3)^5 = a_0 + a_1 x + a_2 x^2 + \cdots + a_{15} x^{15}, then a_{10} equals

If $\sum_{k=0}^{n}\left(\binom{n}{k}\cdot\binom{m}{Ck-Zn}\right) = K\binom{m}{Cn}$, then $K$ is equal to

If $C_0, C_1, C_2, \ldots, C_n$ are binomial coefficients, (where $C_r = ^nC_r$), then the value of $C_0 - C_1 + C_2 - C_3 + \ldots + (-1)^n C_n$ is equal to

If the term independent of x in the expansion of $\left(\sqrt{a} x^{2}+\frac{1}{2 x^{3}}\right)^{10}$ is 105, then $a^{2}$ is equal to:

The coefficient of $x^{-5}$ in the binomial expansion of $\left(\dfrac{x+1}{x^{2/3}-x^{1/3}+1}-\dfrac{x-1}{x-x^{1/2}}\right)^{10}$, where $x\neq 0,1$, is

In the expansion of $(x + a)^n, n \in \mathbb{N}$, if the sum of odd numbered terms be $\alpha$ and the sum of even numbered terms be $\beta$, then:

The value of $2 \times {}^nC_1 + 2^3 \times {}^nC_3 + 2^5 \times {}^nC_5 + \ldots$ is

$^nC_1(1 + \frac{1}{2}){}^nC_2 + (1 + \frac{1}{2} + \frac{1}{3}){}^nC_3 - (1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}){}^nC_4 + .... + (-1)^{n-1}(1 + \frac{1}{2} + \frac{1}{3} + .... + \frac{1}{n}){}^nC_n =$

The value of ${}^{20}C_0 + {}^{20}C_1 + {}^{20}C_2 + {}^{20}C_3 + {}^{20}C_4 + {}^{20}C_{12} + {}^{20}C_{13} + {}^{20}C_{14} + {}^{20}C_{15}$ is

The coefficient of $x^5$ in $(1 + 2x + 3x^2 + \cdots)^{-3/2}$ is

The remainder left out when $8^{2n} - (62)^{2n+1}$ is divided by 9 is

If the coefficients of $r^{\text{th}}$ and $(r+1)^{\text{th}}$ terms in the expansion of $(3+7x)^{29}$ are equal, then $r$ equals

In the expansion of $(1-2\sqrt{x})^{50} = \displaystyle\sum_{r=0}^{50} {}^{50}C_r (1)^{50-r}(-2\sqrt{x})^r = \displaystyle\sum_{r=0}^{50} {}^{50}C_r(-2)^r (x)^{r/2}$, the sum of rational terms is:

$^{16}C_0 - {}^{16}C_1 + {}^{16}C_2 - \ldots + {}^{16}C_8 = 0$. (State whether true or false.)

The number of distinct terms in the expansion of $\left(x + \dfrac{1}{x} + x^2 + \dfrac{1}{x^2}\right)^{15}$ is/are (with respect to different power of $x$)

The term independent of $x$ in the expansion of $\left(1 - \dfrac{1}{x} + 3x^5\right)\left(2x^2 - \dfrac{1}{x}\right)^8$ is:

The remainder when $(27)^{999}$ is divided by 7 is:

Given $\sum_{r=0}^{2n} a_r \left(\frac{1}{x}\right)^r = \left(1 + \frac{1}{x} + \frac{1}{x^2}\right)^n$ and $a_r = a_{2n-r}$ for $0 \le r \le n-1$, find the value of $2(a_0 + a_1 + \ldots + a_{n-1}) + a_n$.

The number of terms in the expansion of $(1 + 5\sqrt{2}x)^9 + (1 - 5\sqrt{2}x)^9$ is

We have $(x + \sqrt{x^3 - 1})^6 + (x - \sqrt{x^3 - 1})^6$. The sum of coefficients of all even powers of $x$ is:

Let $X = ({}^{10}C_1)^2 + 2({}^{10}C_2)^2 + 3({}^{10}C_3)^2 + \ldots + 10({}^{10}C_{10})^2$, where ${}^{10}C_r$, $r \in \{1,2,\ldots,10\}$ denote binomial coefficients. Then the value of $\dfrac{1}{1430}X$ is _______.

For Problems 4–6: The 2nd, 3rd, and 4th terms in the expansion of $(x + a)^n$ are 240, 720, and 1080, respectively.6. The sum of odd-numbered terms is

If$\sum$9 then ($\alpha$+$\beta$) is equal to 9$r+3$9 3 2 ( r ). Cr =$\alpha$( ) -$\beta$,$\alpha$,$\beta$$\ in $N,$r=1$2 2

If the 0th term in the expansion of $\left(\frac{1}{x} + x^2\log x\right)^8$ is 6000, then the value of $x$ is

The coefficient of $x^{10}$ in the expansion of $(1+x)^{10} + (1+x)^{10} + (1+x)^{12} + \ldots + (1+x)^{20}$ is

If $\binom{30}{0}\binom{30}{20}-\binom{30}{1}\binom{30}{19}+\binom{30}{2}\binom{30}{18}-\cdots+\binom{30}{20}\binom{30}{0}={}^nC_r$, then maximum possible value of $n+r$ equals

Find the coefficient of x4 in the expansion of $(1 + x + x^2 + x^3)^{11}$.

For Problems 12–14: Consider the expansion of $(a + b + c + d)^6$. Then the sum of all the coefficients of the terms14. Which contains both $a$ and $b$ is

Consider $f(k) = \dbinom{2k+1}{2k}C - \dbinom{2^k}{2^{k-1}}C$, which of the following is true about $f(k)$

Binomial Theorem Questions (605)