Total number of $3$ letter words that can be formed from the letters of the word $'SAHARANPUR'$ is equal to

Five boys and five girls are writing an examination in a hall in which $5$ benches are arranged in a row and only two students are to be seated on every bench at either of its ends. If the seating arrangement is to be such that no two boys or no two girls sit together as neighbours (a student should not have a student of same gender either on left or right),then the total number of such arrangements is

The number of $5$-digit natural numbers such that the product of their digits is $36$ is:

If $^nP_3 + ^nC_{n-2} = 14n$,then $n = $

In a hotel,four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then the number of all possible ways in which this can be done 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 $\frac{2(x-y)}{10!}=$