Binomial Theorem

The remainder when $(2021)^{2022}+(2022)^{2021}$ is divided by 7 is

The remainder when $7^{2022}+3^{2022}$ is divided by 5 is :

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

Let the number $(22)^{2022}+(2022)^{22}$ leave the remainder $\alpha$ when divided by 3 and $\beta$ when divided by 7. Then $\left(\alpha^{2}+\beta^{2}\right)$ is equal to :

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

Among the statements : (S1) : $2023^{2022}-1999^{2022}$ is divisible by 8 (S2) : $13(13)^{n}-12 n-13$ is divisible by 144 for infinitely many $n \in \mathbb{N}$

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

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

The coefficient of ${x^n}$ in expansion of $\left( {1 + x} \right){\left( {1 - x} \right)^n}$ is

If the coefficient of ${x^7}$ in ${\left[ {a{x^2} + \left( {{1 \over {bx}}} \right)} \right]^{11}}$ equals the coefficient of ${x^{ - 7}}$ in ${\left[ {ax - \left( {{1 \over {b{x^2}}}} \right)} \right]^{11}}$, then $a$ and $b$ satisfy the relation