The number of $4$ letter words (with or without meaning) that can be formed from the eleven letters of the word $'EXAMINATION'$ is

The letters of the word $ASSASSIN$ are written down at random in a row. The probability that no two $S$ occur together is

The word $UNIVERSITY$ is arranged randomly. The probability that both $I$s do not come together is:

There are $10$ pairs of shoes in a cupboard from which $4$ shoes are picked at random. The probability that there is at least one pair is:

Let $0 \leq r \leq n$. If ${ }^{n+1} C_{r+1} : { }^{n} C_{r} : { }^{n-1} C_{r-1} = 55 : 35 : 21$,then $2n + 5r$ is equal to:

Three schools send $2, 4,$ and $6$ students,respectively,to a summer camp. The $12$ students must be accommodated in $6$ rooms numbered $1, 2, 3, 4, 5, 6$ in such a way that each room has exactly $2$ students and both are from the same school. The number of ways the students can be accommodated in the rooms is