In how many ways can a committee consisting of one or more members be formed out of $12$ members of the Municipal Corporation

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

A country has ten smart cities. The government decides to connect all these cities by road. How many roads the government need to construct so that every city is connected to every other city ?

In an examination, a question paper consists of $12$ questions divided into two parts i.e., Part $\mathrm{I}$ and Part $\mathrm{II}$, containing $5$ and $7$ questions, respectively. A student is required to attempt $8$ questions in all, selecting at least $3$ from each part. In how many ways can a student select the questions?

If $^{n + 1}{C_3} = 2{\,^n}{C_2},$ then $n =$

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?

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