Binomial Theorem Questions - JEE

In the expansion of $\left[(1+x)/(1-x)\right]^2$, the coefficient of $x^n$ will be

The largest coefficient in the expansion of $(1 + x)^{24}$ is

Given the binomial expression $\left(\dfrac{x^3}{3} + \dfrac{3}{x}\right)^8$, the middle term is $T_5 = 70x^8$ and $T_5 = 5670$. Find the value of $x$.

The sum of the series $^{20}C_0 + ^{20}C_1 + ^{20}C_2 + \ldots + ^{20}C_{10}$ is

Let P(n) = xn − 1 is divisible by x − k. The least integral value of k is:

If the number of terms in the expansion of $(x + y + z)^n$ are 36, then find the value of $n$.

Coefficient of x^{15} in (1 + x + x^3 + x^4)^n is

If numerically greatest term in the expansion of $(3 - 5x)^{11}$, where $x = \frac{1}{5}$, 729A, then the value of $\frac{a}{b}$ is

The fractional part of $\frac{2^n}{3}$ is

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

The sum of the coefficients of all odd degree terms in the expansion of $\left(x + \sqrt{x^3 - 1}\right)^5 + \left(x - \sqrt{x^3 - 1}\right)^5$, $(x > 1)$ is

For every integer n ≥ 1, which one is correct related to divisibility of $(3^{2^n} - 1)$?

Which term in the expansion of $(2 - 3x)^{19}$ has algebraically the least coefficient?

For Problems 10 and 11: If $a = {}^{20}C_0 - {}^{20}C_3 + {}^{20}C_6 + {}^{20}C_9 + \cdots$; $b = {}^{20}C_1 + {}^{20}C_4 + {}^{20}C_7 + \cdots$; and $c = {}^{20}C_2 - {}^{20}C_5 + {}^{20}C_8 + \cdots$, then11. Value of $(a - b)^2 + (b - c)^2 + (c - a)^2$ is

$3^{51}$ when divided by 8 leaves the remainder

Find the two successive terms in the expansion of $(1 + x)^{24}$ whose coefficients are in the ratio $1 : 4$.

In the expansion of $\left(x + \dfrac{1}{x}\right)^{13}$, every term is a function of $x$. (State whether true or false.)

The number of integral terms in the expansion of $\left(\sqrt{3} + \sqrt[8]{5}\right)^{256}$ is

Find the coefficient of $a^3b^4c^5$ in the expansion of $(bc + ca + ab)^6$.

If $f(x) = \left(x + \dfrac{1}{x}\right)^{2n} + \left(x - \dfrac{1}{x}\right)^{2n}$, then $f(x)$ is a polynomial function which is an even function. (State whether true or false.)

Given $\left(\dfrac{1-t^6}{1-t}\right)^3$, find the coefficient of $t^4$ in its expansion.

If \(|x|

If (1+x)^n = C_0 + C_1 x + C_2 x^2 + C_3 x^3 + \cdots + C_n x^n, n being even, the value of C_0 + (C_0 + C_1) + (C_0 + C_1 + C_2) + \cdots + (C_0 + C_1 + C_2 + \cdots + C_{n-1}) is equal to

If $(1+x)^n = C_0 + C_1x + C_2x^2 + \ldots + C_nx^n$, $n \in N$, then $C_0 - C_1 + C_2 - \ldots + (-1)^{m-1}C_{m-1}$ is equal to \((m

The value of x for which the 6th term in the expansion of \left(\frac{1}{4x} + \frac{1}{x^{2/3}} + 2\right)^n, n ∈ ℕ, is 84, is

Find the sum of possible real values of x for which the sixth term of \left(\frac{1}{7^{\log_7(3|x-2|-9)}} + 7^{\log_3 9|x-2|} \cdot \frac{5}{3}\right)^7 equals 567.

If $^{n+1}C_r \leq (k^2 + 3) \cdot ^nC_{r-1}$, then k belongs to

If the constant term in the expansion of $\left(\dfrac{5\sqrt[3]{x}}{\sqrt{x}}+\dfrac{2x}{\sqrt[3]{5x}}\right)^{12}$, $x\neq0$, is $\alpha\times2^8\times\sqrt[3]{5}$, then $25\alpha$ is equal to:

If xm occurs in the expansion of (x + $\frac{1}{x^{2}}$)2n, then the coefficient of xm is

Given $(x+10)^{50} + (x-10)^{50} = a_0 + a_1 x + a_2 x^2 + \cdots + a_{50}x^{50}$, find the value of $\dfrac{a_2}{a_0}$.

The number of terms in the expansion of \left(\sqrt[5]{x} + \frac{1}{\sqrt[2]{x}}\right)^{8n}, n \in \mathbb{N} is

Let $S_1 = \sum_{j=1}^{10} j(j-1)\,{}^{10}C_j$, $S_2 = \sum_{j=1}^{10} j\,{}^{10}C_j$, and $S_3 = \sum_{j=1}^{10} j^2\,{}^{10}C_j$.Statement 1: $S_3 = 55 \times 2^9$.Statement 2: $S_1 = 90 \times 2^8$ and $S_2 = 10 \times 2^8$.

Integral part of $(7 + 4\sqrt{3})^n$ if $n \in \mathbb{N}$ is

The required coefficient is the coefficient of $x^{12}$ in $(1+x+2x^2)(x+1)^{10}$

$(1+x)^{30} = C_0 + C_1x + C_2x^2 + \ldots + C_{30}x^{30} = \sum_{0}^{30} C_r x^r$

Given that (1+x-2x2)6 = 1 + a1x + a2x2 + ... + a12x12. Find a2 + a4 + a6 + ... + a12.

Let the three consecutive binomial coefficients be ${}^nC_r$, ${}^nC_{r+1}$ and ${}^nC_{r+2}$. Given ${}^nC_r : {}^nC_{r+1} : {}^nC_{r+2} = 2 : 15 : 70$. Find the average of ${}^{16}C_1 + {}^{16}C_2 + {}^{16}C_3$ divided by 3.

The formula $(a + b)^m = a^m + ma^{m-1}b + \dfrac{m(m+1)}{1 \cdot 2} a^{m-2}b^2 + \cdots$ holds when

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

Find the second term after the middle term in the expansion of $\left(\dfrac{x}{2} + \dfrac{2}{x^2}\right)^{20}$.

The value of ${}^nC_0 \times 2^n - {}^nC_1 \times 2^{n-2} + {}^nC_2 \times 2^{n-4}C_r - \ldots$ is equal to

The number of terms in the expansion of $\left(5z + 7z^{-1}\right)^{1224}$ which are integers is

Find the coefficient of i8 in the expansion of $(1 + 2r^2 - r^3)^9$.

If $16^3^{31} - 33$ is divided by 10, then the remainder obtained is

If the 4th term in the expansion of $\left(\sqrt[4]{x^{\log_2 x}} + x^{1/8}\right)$ is 200, then the value of $x$ is (where, $x > -1$)

Given $\displaystyle\sum_{r=0}^{25} {}^{50}C_r \cdot {}^{5-r}C_{25-r} = K \cdot {}^{50}C_{25}$, find the value of $K$.

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

For $n \in \mathbb{N}$, in the expansion of $\left(\sqrt{x^3} + a\sqrt[3]{x^2}\right)^n$, the sum of all the binomial coefficients is between 200 and 400. Also, the term independent of $x$ is 448, then the value of $a$ is

The value of \binom{50}{0} \binom{50}{1} + \binom{50}{1} \binom{50}{2} + \cdots + \binom{50}{49} \binom{50}{50}, where \binom{n}{r} = C_r^n, is

Binomial Theorem Questions (605)