The number of $4$ letter words (with or without meaning) that can be formed from the eleven letters of the word $'EXAMINATION'$ is

If $^{n} C _{9}=\,\,^{n} C _{8},$ find $^{n} C _{17}$

There are $5$ students in class $10,6$ students in class $11$ and $8$ students in class $12.$ If the number of ways, in which $10$ students can be selected from them so as to include at least $2$ students from each class and at most $5$ students from the total $11$ students of class $10$ and $11$ is $100 \mathrm{k}$, then $\mathrm{k}$ is equal to $......$

Consider a class of $5$ girls and $7$ boys. The number of different teams consisting of $2$ girls and $3$ boys that can be formed from this class, if there are two specific boys $A$ and $B$, who refuse to be the members of the same team, is

If $^{2n}{C_3}:{\,^n}{C_2} = 44:3$, then for which of the following values of $r$, the value of $^n{C_r}$ will be 15

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?

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