In how many ways can a girl and a boy be selected from a group of $15$ boys and $8$ girls?

In how many different ways can the letters of the word '$KURUKSHETRA$' be arranged?

There are $12$ members in a group. In how many ways can a team of $6$ members be selected such that a particular member is always included?

$^{14}C_4 + \sum_{j=1}^4 {^{18-j}C_3}$ is equal to

$n$-digit numbers are formed using only three digits $2, 5$,and $7$. The smallest value of $n$ for which $900$ such distinct numbers can be formed is:

There are $m$ points on a straight line $AB$ and $n$ points on another line $AC$,none of them being the point $A$. Triangles are formed from these points as vertices when $(i)$ $A$ is excluded and $(ii)$ $A$ is included. Then the ratio of the number of triangles in the two cases is