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?

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

Let $S=\{1,2,3, \ldots ., 9\}$. For $k=1,2, \ldots \ldots, 5$, let $N_K$ be the number of subsets of $S$, each containing five elements out of which exactly $k$ are odd. Then $N_1+N_2+N_3+N_4+N_5=$

$n$ balls each of weight $w$ when weighted in pairs the sum of the weights of all the possible pairs is $120$ when they are weighed in triplets the sum of the weights comes out to be $480$ for all possible triplets, then $n$ is

$^{14}{C_4} + \sum\limits_{j = 1}^4 {^{18 - j}{C_3}} $ is equal to

Value of $r$ for which $^{15}{C_{r + 3}} = {\,^{15}}{C_{2r - 6}}$ is

Statement$-1:$ The number of ways of distributing $10$ identical balls in $4$ distinct boxes such that no box is empty is $^9C_3 .$

Statement$-2:$ The number of ways of choosing any $3$ places from $9$ different  places is $^9C_3 $.