The total number of three-digit and five-digit integers which can be formed by using the digits $0, 1, 2, 3, 4, 5$,using each digit not more than once in each number,is:

The total number of seven-digit numbers whose sum of digits is even is

There are $3$ applicants for a scholarship in Physics,$5$ for Mathematics,and $4$ for Chemistry. In how many different ways can one of these scholarships be awarded?

The number of integers lying between $1000$ and $10000$ such that the sum of all the digits in each of those numbers becomes $30$ is:

The number of seven-digit integers,with the sum of the digits equal to $10$ and formed by using the digits $1, 2,$ and $3$ only,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 $\frac{2(x-y)}{10!}=$