In how many ways can $12$ gentlemen sit around a round table so that three specified gentlemen are always together?

Let $T_n$ be the number of all possible triangles formed by joining vertices of an $n$-sided regular polygon. If $T_{n+1} - T_n = 10$,then the value of $n$ is:

$A$ code word is to consist of $2$ English alphabets followed by $2$ distinct numbers between $1$ and $9$. For example,$CA23$ is a code word. How many such code words are there?

If $^{2017}C_0 + ^{2017}C_1 + ^{2017}C_2 + ...... + ^{2017}C_{1008} = \lambda^2$ where $\lambda > 0$,then the remainder when $\lambda$ is divided by $33$ is:

The number of ways in which one or more balls can be selected out of $10$ white,$9$ green,and $7$ blue balls are:

Assuming that no two consecutive digits are same,the number of $n$-digit numbers is: