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

Find the number of natural number solutions to the equation $x_1 + x_2 = 100$,such that $x_1$ and $x_2$ are not multiples of $5$.

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:

The sum of all the $4$-digit numbers formed by taking all the digits from ${0, 3, 6, 9}$ without repetition is

The value of $\sum_{r=0}^{20} {}^{50-r}C_{6}$ is equal to

The number of all $3$-digit numbers $abc$ (in base $10$) for which $(a \times b \times c) + (a \times b) + (b \times c) + (c \times a) + a + b + c = 29$ is: