Cardinality of Power Set of a Set

Q.

$\mathrm{S}=\{1,2,3\dots ,20\}$ is to be partitioned into four sets $\mathrm{A},\mathrm{B},\hspace{0.17em}\mathrm{C}$ and $\mathrm{D}$of equal size. The number of ways it can be done, is equal to $?$

1. $\frac{20!}{(4!\mathrm{x}5!)}$

2. $\frac{20!}{{\left(4\right)}^5}$

3. $\frac{20!}{{(5!)}^4}$

4. $\frac{20!}{{(4!)}^5}$

Q.

If a set $A$ has $n$ elements, then the total number of subsets of $A$ is

1. $n$

2. $n^2$

3. $2^n$

4. $2n$

Q.

If A has 3 elements and P(A) denotes the power set of A, then the number of elements in P(P(A)) is

1. 8

2. 9

3. 81

4. 256

Q. Let $Z$ be the set of integers. If
$A=\{x\in Z:2^{(x+2)(x^2-5x+6)}=1\}$ and $B=\{x\in Z:-3<2x-1<9\}$, then the number of subsets of the set $A\times B$, is :

1. $2^{10}$
2. $2^{12}$
3. $2^{15}$
4. $2^{18}$

Q. Let set $R=\{P:B\subseteq P\subseteq A\}.$ If $A=\{1,2,3,4,5\}$ and $B=\{1,2\},$ then the number of elements in set $R$ is

1. $2$
2. $4$
3. $8$
4. $10$

Q. For a set $K,\text{if}n\left(K\right)=m;$ then the number of proper subsets of $K=$

1. $2^m-1$
2. $2^{m-1}$
3. $2^m$