In a party,$12$ persons are to be arranged around a round table. If two particular persons are not to be seated side by side,then the total number of arrangements is

An eight-digit number divisible by $9$ is to be formed using digits from $0$ to $9$ without repeating the digits. The number of ways in which this can be done is: (in $(7!)$)

There are $6$ multiple choice questions in an examination. How many sequences of answers are possible,if the first three questions have $4$ choices each and the next three have $5$ choices each?

In how many ways can a cricket team of $11$ players be selected out of $16$ players if $1$ particular player is to be excluded?

The number of ways in which the letters of the word $TRIANGLE$ can be arranged such that no two vowels occur together 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