The number of arrangements of the letters of the word $SALOON$,if the two $O$'s do not come together,is

If $^nP_5 = 9 \times ^{n-1}P_4$,then the value of $n$ is

In how many ways can a team of $10$ players be selected from $22$ players if $6$ particular players are always to be included and $4$ particular players are always excluded?

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

In a plane there are $37$ straight lines of which $13$ pass through the point $A$ and $11$ pass through the point $B$. Besides,no three lines pass through one point,no line passes through both points $A$ and $B$,and no two lines are parallel. Then the number of intersection points the lines have is equal to

Let $n$ be the number of different $5$-digit numbers,divisible by $4$,formed with the digits $1, 2, 3, 4, 5,$ and $6$,such that no digit is repeated in the numbers. What is the value of $n$?