How many numbers lying between $100$ and $1000$ can be formed with the digits $0, 1, 2, 3, 4, 5$,if the repetition of the digits is not allowed?

The number of five-digit numbers divisible by $5$ that can be formed using the digits $0, 1, 2, 3, 4, 5$ without repetition is

In a class of $20$ students,in how many ways can the following prizes be awarded?
If the first and second prizes are for Mathematics,the first and second prizes for Physics,the first prize for Chemistry,and the first prize for English are to be awarded.

The number of different nine-digit numbers that can be formed by rearranging the digits of the number $223355888$ such that the odd digits occupy even positions is:

How many arrangements of $3$ letters can be made from the letters $a, b, c, d, e, f$ without repetition,such that each arrangement contains at least one vowel?

The domain of the function $f(x) = {}^{7-x}P_{x-1}$ is: