Binomial Theorem

Sum of the last 30 coefficients in the expansion of $(1+x)^{59}$, when expanded in ascending power of $x$ is

The number of positive integers satisfying the inequality ${ }^{n+1} C_{n-2}-{ }^{n+1} C_{n-1} \leq 50$ is

The remainder obtained when $1!+2!+\ldots+200!$ is divided by 14 is

The least remainder when $17^{30}$ is divided by 5 , is

Given positive integers $r\gt1, n\gt2$ and the coefficient of (3r) th and $(r+2)$ th terms in the binomial expansion of $(1+x)^{2 n}$ are equal, then

A person is permitted to select at least one and at most $n$ coins from a collection of $(2 n+1)$ distinct coins. It the total number of ways in which he can select coins is 255 , then $n$ equals

The remainder obtained when $5^{124}$ is divided by 124 is

The least positive remainder when $123 \times 125 \times 127$ is divided by 124 is

The statement $P(n)=9^n-8^n$, when divided 8 , always leaves the remainder

Last two digits in $(9)^{50}$ be____

The value of $\frac{C(n, 2)}{(n+1)!}$ is

If $t_n=\sum_{r=0}^n \frac{1}{\left({ }^n C_r\right)^k}$ and $S_n=\sum_{r=0}^n \frac{r}{\left({ }^n C_r\right)^k}$, where $k \in Z^{+}$, then $\cos ^{-1}\left(\frac{S_n}{n t_n}\right)$ is