Which is larger ${99}^{50}+{100}^{50}$ or ${\left(101\right)}^{50}$?

Important Questions on Binomial Theorem

HARD

Mathematics>Algebra>Binomial Theorem>General Term in Binomial Expansion

Let $\mathrm{X}$ be a set containing $10$ elements and $\mathrm{P}\left(\mathrm{X}\right)$ be its power set. If $\mathrm{A}$ and $\mathrm{B}$ are picked up at random from $\mathrm{P}\left(\mathrm{X}\right)$, with replacement, then the probability that $A$ and $\mathrm{B}$ have equal number of elements is:

HARD

Mathematics>Algebra>Binomial Theorem>General Term in Binomial Expansion

Let $X={\left({}^{10}{C}_1\right)}^2+2{\left({}^{10}{C}_2\right)}^2+3{\left({}^{10}{C}_3\right)}^2+\dots +10{\left({}^{10}{C}_{10}\right)}^2$ , where ${}^{10}{C}_r, r\in \left\{\mathrm{1,2},\dots .,10\right\}$ denote binomial coefficients. Then, the value of $\frac{1}{1430}X$ is ______ .

MEDIUM

Mathematics>Algebra>Binomial Theorem>General Term in Binomial Expansion

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

MEDIUM

Mathematics>Algebra>Binomial Theorem>General Term in Binomial Expansion

Let $x_n={\left(2^n+3^n\right)}^{\frac{1}{2n}}$ for all natural numbers $n$. Then

MEDIUM

Mathematics>Algebra>Binomial Theorem>General Term in Binomial Expansion

If $\sum _{r=0}^{25}\left\{\left({}_{ }{}^{50}C_r\right)\left({}_{ }{}^{50-r}C_{25-r}\right)\right\}=K\left({}_{ }{}^{50}C_{25}\right)$, then $K$ is equal to

MEDIUM

Mathematics>Algebra>Binomial Theorem>General Term in Binomial Expansion

If the coefficient of the three successive terms in the binomial expansion of ${\left(1+x\right)}^n$ are in the ratio $1:7:42$, then the first of these terms in the expansion is

HARD

Mathematics>Algebra>Binomial Theorem>General Term in Binomial Expansion

The term independent of $x$ in the expansion of ${\left(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}\right)}^{10}$  is

MEDIUM

Mathematics>Algebra>Binomial Theorem>General Term in Binomial Expansion

The value of $\left({}^{21}{C}_1{-}^{10}{C}_1\right)+\left({}^{21}{C}_2{-}^{10}{C}_2\right)+\left({}^{21}{C}_3{-}^{10}{C}_3\right)+\left({}^{21}{C}_4{-}^{10}{C}_4\right)+\dots +\left({}^{21}{C}_{10}{-}^{10}{C}_{10}\right)$is

MEDIUM

Mathematics>Algebra>Binomial Theorem>General Term in Binomial Expansion

If$f\left(x\right) = x^n$, then the value of  $f\left(1\right)-\frac{f^{\text{'}}\left(1\right)}{1!}+\frac{f^{\text{'}\text{'}}\left(1\right)}{2!}-\frac{f^{\text{'}\text{'}\text{'}}\left(1\right)}{3!}+\dots +\frac{{\left(-1\right)}^n{f}^n\left(1\right)}{n!}$  is

HARD

Mathematics>Algebra>Binomial Theorem>General Term in Binomial Expansion

The total number of irrational terms in the binomial expansion of ${\left(7^{\frac{1}{5}}-3^{\frac{1}{10}}\right)}^{60}$ is

HARD

Mathematics>Algebra>Binomial Theorem>General Term in Binomial Expansion

The fractional part of a real number $x$ is $x-\left[x\right],$ where $\left[x\right]$ is the greatest integer less than or equal to $x.$ Let $F_1$ and $F_2$ be the fractional parts of ${\left(44-\sqrt{2017}\right)}^{2017}$ and ${\left(44+\sqrt{2017}\right)}^{2017}$ respectively. Then $F_1+F_2$ lies between the numbers

HARD

Mathematics>Algebra>Binomial Theorem>General Term in Binomial Expansion

If the third term in the binomial expansion of ${\left(1+x^{{\log }_{2}x}\right)}^5$ equals $2560,$ then a possible value of $x$ is

MEDIUM

Mathematics>Algebra>Binomial Theorem>General Term in Binomial Expansion

The digit in the unit place of $7^{291}$ is

HARD

Mathematics>Algebra>Binomial Theorem>General Term in Binomial Expansion

If in a regular polygon the number of diagonals is $54$, then the number of sides of this polygon is:

HARD

Mathematics>Algebra>Binomial Theorem>General Term in Binomial Expansion

The coefficient of $x^2$ in the expansion of the product $\left(2-x^2\right)\left\{{\left(1+2x+3x^2\right)}^6+{\left(1-4x^2\right)}^6\right\}$ is

HARD

Mathematics>Algebra>Binomial Theorem>General Term in Binomial Expansion

${\left(\sqrt{3}+\sqrt{2}\right)}^4-{\left(\sqrt{3}-\sqrt{2}\right)}^4=$

MEDIUM

Mathematics>Algebra>Binomial Theorem>General Term in Binomial Expansion

If the sum of the coefficients in the expansions of ${\left(a^2{x}^2-2ax+1\right)}^{51}$ is zero, then $a$ is equal to

MEDIUM

Mathematics>Algebra>Binomial Theorem>General Term in Binomial Expansion

The value of $\left[C\left(7,0\right)+C\left(7,1\right)\right]+\left[C\left(7,1\right)+C\left(7,2\right)\right]+\dots +\left[C\left(7,6\right)+C\left(7,7\right)\right]$ is

MEDIUM

Mathematics>Algebra>Binomial Theorem>General Term in Binomial Expansion

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

HARD

Mathematics>Algebra>Binomial Theorem>General Term in Binomial Expansion

The sum of coefficients of integral powers of $x$ in the binomial expansion of ${\left(1-2\sqrt{x}\right)}^{\text{50}}$ is