The number of times the digit $5$ will be written when listing the integers from $1$ to $1000$ is

Let $A = \{a_1, a_2, a_3, ..., a_n\}$ be a set containing $n$ elements. Two subsets $P$ and $Q$ of $A$ are formed independently. The number of ways in which these subsets can be formed such that $(P - Q)$ contains exactly $2$ elements is:

$A$ committee of $5$ members is to be formed out of $3$ trainees,$4$ professors,and $6$ research associates. In how many different ways can this be done if the committee should have all $4$ professors and $1$ research associate $OR$ all $3$ trainees and $2$ professors?

How many different signals can be transmitted by arranging $3$ red,$2$ yellow and $2$ green flags on a pole? [Assume that all the $7$ flags are used to transmit a signal.]

The total number of words that can be formed by using all the letters of the word $ALLEN$ such that all vowels occur in alphabetical order is:

In how many ways,a committee of $3$ members can be selected from $5$ men and $4$ women,consisting of $2$ men and $1$ woman?