In how many ways can four children be made to stand in a line such that two of them,$A$ and $B$,are always together?

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

How many words of $4$ consonants and $3$ vowels can be formed from $6$ consonants and $5$ vowels?

$A$ committee of $5$ members is to be formed out of $3$ trainees,$4$ professors,and $6$ research associates. In how many different ways can this be done if the committee should have all $4$ professors and $1$ research associate $OR$ all $3$ trainees and $2$ professors?

The number of ways in which $3$ children can distribute $10$ tickets out of $15$ consecutively numbered tickets among themselves such that they get consecutive blocks of $5, 3$ and $2$ tickets is:

If ${ }^{2 n} C_{r} = { }^{2 n} C_{r+2}$,find $r$.