Binomial Theorem Questions - JEE

If 1 2$\cdot$( 15 C 1$) + 2$2$\cdot$( 15 C 2$) + 3$2$\cdot$( 15 C 3 ) +$\ldots$+15 2$\cdot$( 15 C 15$) = 2$m$\cdot$3 n$\cdot$5 k , where m, n, k$\ in $N, then$m + n + k$is equal to :

The sum of the series 2 $\times$ 1 $\times$ 20$C_{4}$- 3 $\times$ 2 $\times$ 20$C_{5}$+ 4 $\times$ 3 $\times$ 20$C_{6}$- 5 $\times$ 4 $\times$ 20$C_{7}$+$\ldots$+ 18 $\times$ 17 $\times$ 20$C_{20}$, is equal to

The sum of all rational terms in the expansion of (2 +$\sqrt{3}$) is 8

For an integer n$\ge$2, if the arithmetic mean of all coefficients in the binomial expansion of$(x + y)$$2n-3$is 16 , then the distance of the point P$(2n - 1$,$n - 4n)$from the line$x + y = 8$is: 2

The integral part of $(8 + 3\sqrt{7})^{20}$ is even. (State whether true or false.)

The number of terms in the expansion of $(1 + 2x + x^2)^n$ when expanded in descending powers of $x$, is

Determine the term independent of $a$ in the expansion of $\left(\dfrac{a+1}{a^{2/3} - a^{1/3} + 1} - \dfrac{a-1}{a - a^{1/2}}\right)^{10}$.

Find the coefficient of $a^6 b^6$ in the expansion of $\left(a^2 - \dfrac{b}{a}\right)^{12}$.

If $a_n = \sum_{r=0}^{n} \frac{1}{C_r}$, then $\sum_{r=0}^{n} \frac{r^2}{C_r} = P(n)a_{n+2} + Q(n)a_{n+1} + a_n + R(n)$, where $P(n), Q(n), R(n)$ are the polynomial functions of $n$, then $R(5) =$

If $\left(x - 2 + \dfrac{1}{x}\right)^{30} = a_0 x^{30} + a_1 x^{29} + \ldots + a_{29}x + a_{30} x^{-1} + \ldots + a_{60}x^{-30}$ and $k = a_0 + a_1 + \ldots + a_{60}$. If $k - a_{30} = -{}^nC_r$, then $n + r$ is equal to

Coefficient of x5 in the expansion of (1 + x + x2)15 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

If $(1+x+x^2)^n=\sum_{r=0}^{2n}a_r x^r$, then $a_r-{}^nC_1 a_{r-1}+{}^nC_2 a_{r-2}-\cdots+(-1)^r{}^nC_r$ equals (r not multiple of 3)

The sum of the rational terms in the binomial expansion of $\left(2^{\frac{1}{2}}+3^{\frac{1}{5}}\right)^{10}$ is:

For $\forall$n $\in$ N, 23n+3 - 8 is divisible by:

The ratio of the 5th term from the beginning to the 5th term from the end in the binomial expansion of $\left(2^{\frac{1}{3}}+\frac{1}{2(3)^{\frac{1}{3}}}\right)^{10}$is

The coefficient of $x^{256}$ in the expansion of $(1-x)^{101}\left(x^{2}+x+1\right)^{100}$ is:

If the coefficient of $x^7$ in the expansion of $\left(ax^2 + \frac{1}{bx}\right)^{11}$ and coefficient of $x^{-7}$ in the expansion of $\left(ax - \frac{1}{bx^2}\right)^{11}$ are equal, then the minimum value of $a^2 + b^2$ is

Match each binomial sum in List-I with the minimum value of $p+q+m+n$ (where the sum $=\binom{p}{q}\cdot\binom{n}{m}$): P) $\displaystyle\sum_{r=0}^{10}\binom{25}{r}\binom{10}{r}\binom{r+10}{r}$; Q) $\displaystyle\sum_{r=0}^{10}\binom{15}{10-r}\binom{25}{r}\binom{25-r}{10-r}$; R) $\displaystyle\sum_{r=0}^{16}(-1)^r\binom{25}{r}\binom{25-r}{9}\binom{16}{r}$; S) $\displaystyle\sum_{r=0}^{18}(-1)^r\binom{28}{r}\binom{r}{10}\binom{18}{r}$ List-II: 1)58; 2)60; 3)85; 4)70; 5)110

If the second, third and fourth terms in the expansion of $(x+y)^n$ are 135, 30 and $\dfrac{10}{3}$, respectively, then $6(n^3+x^2+y)$ is equal to _______.

Let the integral part of (8 + $3 \sqrt{7}$)n = 1, then

If x = 1 + $\frac{3}{1 !} \times \frac{1}{6}+\frac{3 \times 7}{2 !}\left(\frac{1}{6}\right)^{2}+\frac{3 \times 7 \times 11}{3 !}\left(\frac{1}{6}\right)^{3}$ + ... then x4 =

If $\left(x - 2 + \dfrac{1}{x}\right)^{30} = a_0 x^{30} + a_1 x^{29} + \ldots + a_{29}x + a_{30} x^{-1} + \ldots + a_{60}x^{-30}$ and $k = a_0 + a_1 + \ldots + a_{60}$. If $k - a_{30} = -{}^nC_r$, then $n + r$ is equal to

If (1 + x)n = C0 + C1x + C2x2 + ... + Cnxn, then the value of C0 + 2C1 + 3C2 + ... + (n + 1)Cn is

The expression $\left(x+\sqrt{x^{2}-1}\right)^{5}+\left(x-\sqrt{x^{2}-1}\right)^{5}$ is a polynomial of degree

If in the expansion of $(1 + z)^m(1 - z)^n$ the coefficients of $z$ and $z^2$ are $-6$ and $-6$ respectively, then the value of $m + n$ is

If $\frac{1}{m} + \frac{2}{m} + \frac{2^2}{m^2} + \ldots + \frac{2^n}{m^n} = \frac{s}{7}$, then the value of $m + n$ is

The sum of the coefficients of three consecutive terms in the binomial expansion of $(1+x)^{n+2}$, which are in the ratio $1:3:5$, is equal to

Fractional part of the number $\dfrac{4^{2022}}{15}$ is equal to

Given $ \displaystyle\sum_{i=1}^{20} \left(\dfrac{{}^{20}C_{i-1}}{{}^{20}C_i + {}^{20}C_{i-1}}\right)^3 = \dfrac{k}{21} $, find the value of $ k $.

The coefficient of $x^8$ in the expansion of $\left(1 + \frac{x}{2} + \frac{x^2}{4} + \frac{x^3}{8} + \ldots\right)^{10}$ is

If $(1+x)^{2010}=C_0+C_1x+\cdots+C_{2010}x^{2010}$, then $C_2+C_5+C_8+\cdots+C_{2009}$ equals

$25^{190}-19^{190}-8^{190}+2^{190}$ is divisible by

$^nC_m - 3^{n-1}C_m + 5^{n-2}C_m - 7^{n-3}C_m + \ldots + (2(n-m)-1)^nC_m$ is equal to:

$25^{190}-19^{190}-8^{190}+2^{190}$ is divisible by

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 $\sqrt{6}:1$, then the third term from the beginning is:

If $n \in \mathbb{N}$ and $(1 + x + x^2)^n = \sum_{r=0}^{2n} a_r x^r$, then $\sum_{r=0}^{n} (-1)^r a_r \, ^nC_r$ is equal to:

Let $k$ and $n$ be the positive integers and $S_k = 1^k + 2^k + 3^k + .... + n^k$. Then $^{n+1}C_{k+1} + ^{n+1}C_2S_2 + ^{n+1}C_3S_3 + .... + ^{n+1}C_mS_m$ is equal to:

Let $f_1(x) = (x-2)^2$, $f_2(x) = ((x-2)^2 - 2)^2$, $f_3(x) = \left((x-2)^2 - 2)^2 - 2\right)^2$, and so on; so that $f_k(x) = \left(...\left((x-2)^2 - 2\right)^2 - ... - 2\right)^2 = A_k + B_k x + C_k x^2 + D_k x^3 + ...$. Then $C_3$ is equal to:

Among the statements: (S1): $2023^{2022}-1999^{2022}$ is divisible by $8$. (S2): $13(13)^n-11n-13$ is divisible by $144$ for infinitely many $n\in\mathbb{N}$.

$\sum_{r=1}^{n} \frac{(-1)^{r-1}{}^nC_r 0 - y^f}{r} =$

If the unit digit of $13^n + 7^n - 3^n$, $n \in \mathbb{N}$, is 3 then possible value(s) of $n$ is/are :

If the coefficient of $x^t$ and $x^{t+1}$ in $\sum_{r=0}^{n}(1+x)^r$ where $t < n-2$ are equal, then :

The value of $\sum_{i=1}^{n}\frac{^nC_i}{i}$ is equal to

If the coefficient of $x^{15}$ in the expansion of $\left(ax^3+\dfrac{1}{bx^{1/3}}\right)^{15}$ is equal to the coefficient of $x^{-15}$ in the expansion of $\left(ax^{1/3}-\dfrac{1}{bx^3}\right)^{15}$, where $a$ and $b$ are positive real numbers, then for each such ordered pair $(a,b)$:

If $a_n = \sum_{r=0}^{n} \frac{1}{C_r}$, then $\sum_{r=0}^{n} \frac{r^2}{C_r} = P(n)a_{n+2} + Q(n)a_{n+1} + a_n + R(n)$, where $P(n), Q(n), R(n)$ are the polynomial functions of $n$, then $P(5) =$

Let $K$ be the sum of the coefficients of the odd powers of $x$ in the expansion of $(1+x)^{99}$. Let $a$ be the middle term in the expansion of $\left(2+\dfrac{1}{\sqrt{2}}\right)^{200}$. If $\dfrac{{}^{200}C_{99}\,K}{a}=\dfrac{2^\ell\,m}{n}$, where $m$ and $n$ are odd numbers, then the ordered pair $(\ell,n)$ is equal to:

$(1+x)^n = \sum_{r=0}^{n} C_r x^r$

Let $n \in \mathbb{N}, n \geq 4$ and $P = \prod_{r=0}^{n} \, ^nC_r$, then:

Binomial Theorem Questions (605)