Let $m$ (respectively,$n$) be the number of $5$-digit integers obtained by using the digits $1, 2, 3, 4, 5$ with repetitions (respectively,without repetitions) such that the sum of any two adjacent digits is odd. Then $\frac{m}{n}$ is equal to

In a high school,a committee has to be formed from a group of $6$ boys $M_1, M_2, M_3, M_4, M_5, M_6$ and $5$ girls $G_1, G_2, G_3, G_4, G_5$.
$(i)$ Let $\alpha_1$ be the total number of ways in which the committee can be formed such that the committee has $5$ members,having exactly $3$ boys and $2$ girls.
$(ii)$ Let $\alpha_2$ be the total number of ways in which the committee can be formed such that the committee has at least $2$ members,and having an equal number of boys and girls.
$(iii)$ Let $\alpha_3$ be the total number of ways in which the committee can be formed such that the committee has $5$ members,at least $2$ of them being girls.
$(iv)$ Let $\alpha_4$ be the total number of ways in which the committee can be formed such that the committee has $4$ members,having at least $2$ girls and such that both $M_1$ and $G_1$ are $NOT$ in the committee together.

$LIST-I$	$LIST-II$
$P$. The value of $\alpha_1$ is	$1. 136$
$Q$. The value of $\alpha_2$ is	$2. 189$
$R$. The value of $\alpha_3$ is	$3. 192$
$S$. The value of $\alpha_4$ is	$4. 200$
	$5. 381$
	$6. 461$

The correct option is:

Five boys and five girls are writing an examination in a hall in which $5$ benches are arranged in a row and only two students are to be seated on every bench at either of its ends. If the seating arrangement is to be such that no two boys or no two girls sit together as neighbours (a student should not have a student of same gender either on left or right),then the total number of such arrangements is

Let $a_1, a_2, \ldots, a_n$ be $n$ non-zero real numbers,of which $p$ are positive and the remaining are negative. The number of ordered pairs $(j, k)$ with $j < k$ for which $a_j a_k$ is positive is $55$. Similarly,the number of ordered pairs $(j, k)$ with $j < k$ for which $a_j a_k$ is negative is $50$. Then,the value of $p^2 + (n-p)^2$ is

The number of times the digit $3$ will be written when listing the integers from $1$ to $1000$ is

The letters of the word $ASSASSIN$ are written down at random in a row. The probability that no two $S$ occur together is