The total number of terms in the expansion of $(x+a)^{47}-(x-a)^{47}$ after simplification is

If $x$ is so small that $x^2$ and higher powers of $x$ may be neglected,then the approximate value of $\frac{(1+\frac{2}{3}x)^{-3}(1-15x)^{-1/5}}{(2-3x)^4}$ is:

For natural numbers $m, n$,if $(1 - y)^m(1 + y)^n = 1 + a_1y + a_2y^2 + \ldots$ and $a_1 = a_2 = 10$,then $(m, n) = \_\_\_\_\_\_$.

If $f(x)=(2011+x)^n$,where $x$ is a real variable and $n$ is a positive integer,then the value of $f(0)+f^{\prime}(0)+\frac{f^{\prime \prime}(0)}{2 !}+\ldots+\frac{f^{(n-1)}(0)}{(n-1) !}$ is

Let $n \geq 5$ be an integer. If $9^{n}-8n-1=64\alpha$ and $6^{n}-5n-1=25\beta$,then $\alpha-\beta$ is equal to

Using the binomial theorem, the value of $(0.999)^3$ correct to $3$ decimal places is: