The coefficient of $x^{53}$ in the expansion $\sum_{m = 0}^{100} {^{100}C_m (x - 3)^{100 - m} \cdot 2^m}$ is

If $n$ is an even positive integer,then the condition that the greatest term in the expansion of $(a+x)^{n}$ may also have the greatest coefficient is

$p, q$ are two prime numbers. For $n=pq$,if the expansion $\left(x^{-5/4} + 2x^{4/5}\right)^n$ contains non-zero coefficients of $x^{-n}$ and $x^0$,then the least value of such $n$ is

The ratio of the coefficient of $x^2$ to the coefficient of $x^{10}$ in the expansion of $(x^5 + 4 \cdot 3^{-\log_{\sqrt{3}}\sqrt{x^3}})^{10}$ is

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 $(2+\frac{1}{\sqrt{2}})^{200}$. If $\frac{{}^{200}C_{99} K}{a} = \frac{2^{\ell} m}{n}$,where $m$ and $n$ are odd numbers,then the ordered pair $(\ell, n)$ is equal to:

In the expansion of $(1 + x)^{43}$,if the coefficients of the $(2r + 1)^{th}$ and the $(r + 2)^{th}$ terms are equal,the value of $r$ is: