All possible two factors products are formed from numbers $1, 2, 3, 4, ...., 200$. The number of factors out of the total obtained which are multiples of $5$ is

In how many ways can $6$ persons be selected from $4$ officers and $8$ constables, if at least one officer is to be included

The number of four-letter words that can be formed with letters $a, b, c$ such that all three letters occur is

The total number of natural numbers of six digits that can be made with digits $1, 2, 3, 4$, if all digits are to appear in the same number at least once, is

Let

$S _1=\{( i , j , k ): i , j , k \in\{1,2, \ldots, 10\}\}$

$S _2=\{( i , j ): 1 \leq i < j +2 \leq 10, i , j \in\{1,2, \ldots, 10\}\},$

$S _3=\{( i , j , k , l): 1 \leq i < j < k < l, i , j , k , l \in\{1,2, \ldots ., 10\}\}$

$S _4=\{( i , j , k , l): i , j , k$ and $l$ are distinct elements in $\{1,2, \ldots, 10\}\}$

and  If the total number of elements in the set $S _t$ is $n _z, r =1,2,3,4$, then which of the following statements is (are) TRUE?

$(A)$ $n _1=1000$   $(B)$ $n _2=44$   $(C)$ $n _3=220$   $(D)$ $\frac{ n _4}{12}=420$

The total number of three-digit numbers, with one digit repeated exactly two times, is