Ten students are seated at random in a row. The probability that two particular students are not seated side by side is

In a group of $3$ girls and $4$ boys,there are two boys $B_1$ and $B_2$. The number of ways in which these girls and boys can stand in a queue such that all the girls stand together,all the boys stand together,but $B_1$ and $B_2$ are not adjacent to each other,is:

$A$ group of students comprises $5$ boys and $n$ girls. If the number of ways,in which a team of $3$ students can be randomly selected from this group such that there is at least one boy and at least one girl in each team,is $1750$,then $n$ is equal to

Words of length $10$ are formed by using the letters $A, B, C, D, E, F, G, H, I, J$. Let $x$ be the number of such words where no letter is repeated and $y$ be the number of such words where exactly two letters are repeated twice and no other letter is repeated,then the value of $\frac{y}{x}$ is

The number of $4$-digit numbers that can be formed from the digits $0, 1, 2, 3, 4, 5, 6, 7$ such that each number contains the digit $1$ is:

$A$ five-digit number divisible by $3$ has to be formed using the numerals $0, 1, 2, 3, 4,$ and $5$ without repetition. The total number of ways in which this can be done is: