If a seven-digit number formed with distinct digits $4, 6, 9, 5, 3, x,$ and $y$ is divisible by $3$,then the number of such ordered pairs $(x, y)$ is

There are $10$ red and $5$ yellow roses of different sizes. If $x$ is the number of garlands that can be formed with all these flowers so that no two yellow roses come together and $y$ is the number of garlands formed with all these flowers so that all the red roses come together,then $\frac{2(x-y)}{10!}=$

How many words can be formed by taking two identical and two distinct letters from the letters of the word $COMBINATION$?

The numbers $1, 2, 3, \ldots, n$ are arranged in a random order. The probability that the digits $1, 2, 3, \ldots, k$ appear as a block in that order is:

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

If ${}^n C_{r-1}=36$,${}^n C_r=84$,and ${}^n C_{r+1}=126$,then the value of ${}^n C_8$ is