If ${ }^{1} P_{1}+2 \cdot{ }^{2} P_{2}+3 \cdot{ }^{3} P_{3}+\ldots+15 \cdot{ }^{15} P_{15}={ }^{q} P_{r}-s$,where $0 \leq s \leq 1$,then ${ }^{q+s} C_{r-s}$ is equal to .... .

The number of $3$-digit numbers,formed using the digits $2, 3, 4, 5$ and $7$,when the repetition of digits is not allowed,and which are not divisible by $3$,is equal to ..........

How many $5$-digit numbers can be formed using the digits $0, 1, 2, 3, 4,$ and $5$ without repetition such that the number is divisible by $3$?

In an admission test,there are $15$ multiple-choice questions. Each question has $4$ alternatives. For each question,one or more than one answer can be correct. If a student attempts all $15$ questions and marks the answers randomly,in how many different ways can the student answer the question paper?

For which value of $n \in N$,does $n!$ have $13$ trailing zeros?

Let $x$ denote the number of ways of arranging $m$ boys and $m$ girls in a row so that no two boys sit together. If $y$ and $z$ denote the number of ways of arranging $m$ boys and $m$ girls in a row and around a circular table respectively so that boys and girls sit alternately,then $x: y: z=$