Let $S = \{ 1, 2, \ldots, n \}$. The number of possible pairs of the form $(A, B)$ with $A \subseteq B$ for subsets $A$ and $B$ of $S$ is
(A) $2^n$
(B) $3^n$
(C) $\sum _ { k = 0 } ^ { n } \binom { n } { k } \binom { n } { n - k }$
(D) $n!$

Let $S = \{ 1,2 , \ldots , n \}$. Find the number of unordered pairs $\{ A , B \}$ of subsets of $S$ such that $A$ and $B$ are disjoint, where $A$ or $B$ or both may be empty.

Let $S = \{ 1,2 , \ldots , n \}$. Find the number of unordered pairs $\{ A , B \}$ of subsets of $S$ such that $A$ and $B$ are disjoint, where $A$ or $B$ or both may be empty.

Let $S = \{ 1, 2, \ldots, n \}$. The number of possible pairs of the form $(A, B)$ with $A \subseteq B$ for subsets $A$ and $B$ of $S$ is
(A) $2 ^ { n }$
(B) $3 ^ { n }$
(C) $\sum _ { k = 0 } ^ { n } \binom { n } { k } \binom { n } { n - k }$
(D) $n !$

Let $S = \{ 1, 2, \ldots , n \}$. The number of possible pairs of the form $(A, B)$ with $A \subseteq B$ for subsets $A$ and $B$ of $S$ is
(A) $2 ^ { n }$
(B) $3 ^ { n }$
(C) $\sum _ { k = 0 } ^ { n } \binom { n } { k } \binom { n } { n - k }$
(D) $n !$

The number of subsets of $\{ 1,2,3 , \ldots , 10 \}$ having an odd number of elements is
(A) 1024
(B) 512
(C) 256
(D) 50 .

Let $S = \{ 1,2 , \ldots , n \}$. For any non-empty subset $A$ of $S$, let $l ( A )$ denote the largest number in $A$. If $f ( n ) = \sum _ { A \subseteq S } l ( A )$, that is, $f ( n )$ is the sum of the numbers $l ( A )$ while $A$ ranges over all the nonempty subsets of $S$, then $f ( n )$ is
(A) $2 ^ { n } ( n + 1 )$
(B) $2 ^ { n } ( n + 1 ) - 1$
(C) $2 ^ { n } ( n - 1 )$
(D) $2 ^ { n } ( n - 1 ) + 1$.

Let $X$ be the set $\{ 1,2,3 , \ldots , 10 \}$ and $P$ the subset $\{ 1,2,3,4,5 \}$. The number of subsets $Q$ of $X$ such that $P \cap Q = \{ 3 \}$ is
(A) 1
(B) $2 ^ { 4 }$
(C) $2 ^ { 5 }$
(D) $2 ^ { 9 }$

Let $S = \{ 1,2,3,4 \}$. The total number of unordered pairs of disjoint subsets of $S$ is equal to
A) 25
B) 34
C) 42
D) 41

Let $S = \{ 1,2,3,4,5,6 \}$ and $X$ be the set of all relations $R$ from $S$ to $S$ that satisfy both the following properties: i. $R$ has exactly 6 elements. ii. For each $( a , b ) \in R$, we have $| a - b | \geq 2$. Let $Y = \{ R \in X$ : The range of $R$ has exactly one element $\}$ and $Z = \{ R \in X : R$ is a function from $S$ to $S \}$. Let $n ( A )$ denote the number of elements in a set $A$. If $n ( X ) = {}^{ m } C _ { 6 }$, then the value of $m$ is $\_\_\_\_$ .

Let $S = \{ 1,2,3,4,5,6 \}$ and $X$ be the set of all relations $R$ from $S$ to $S$ that satisfy both the following properties: i. $R$ has exactly 6 elements. ii. For each $( a , b ) \in R$, we have $| a - b | \geq 2$. Let $Y = \{ R \in X$ : The range of $R$ has exactly one element $\}$ and $Z = \{ R \in X : R$ is a function from $S$ to $S \}$. Let $n ( A )$ denote the number of elements in a set $A$. If the value of $n ( Y ) + n ( Z )$ is $k ^ { 2 }$, then $| k |$ is $\_\_\_\_$ .

Let $A$ and $B$ be two sets containing 2 elements and 4 elements respectively. The number of subsets of $A \times B$ having 3 or more elements is:
(1) 219
(2) 211
(3) 256
(4) 220

Let $A$ and $B$ be two sets containing four and two elements respectively. Then the number of subsets of the set $A \times B$, each having at least three elements is
(1) 510
(2) 219
(3) 256
(4) 275

Let $S = \{ 1,2,3 , \ldots , 100 \}$, then number of non-empty subsets $A$ of $S$ such that the product of elements in $A$ is even is :
(1) $2 ^ { 100 } - 1$
(2) $2 ^ { 50 } + 1$
(3) $2 ^ { 50 } \left( 2 ^ { 50 } - 1 \right)$
(4) $2 ^ { 50 } - 1$

Let $Z$ be the set of all integers, $A = \left\{ ( x , y ) \in Z \times Z : ( x - 2 ) ^ { 2 } + y ^ { 2 } \leq 4 \right\}$ $B = \left\{ ( x , y ) \in Z \times Z : x ^ { 2 } + y ^ { 2 } \leq 4 \right\}$ and $C = \left\{ ( x , y ) \in Z \times Z : ( x - 2 ) ^ { 2 } + ( y - 2 ) ^ { 2 } \leq 4 \right\}$ If the total number of relations from $A \cap B$ to $A \cap C$ is $2 ^ { p }$, then the value of $p$ is: (1) 25 (2) 9 (3) 16 (4) 49

Let the number of elements in sets $A$ and $B$ be five and two respectively. Then the number of subsets of $A \times B$ each having at least 3 and at most 6 elements is
(1) 752
(2) 782
(3) 792
(4) 772

Let $A$ and $B$ be two finite sets with $m$ and $n$ elements respectively. The total number of subsets of the set $A$ is 56 more than the total number of subsets of $B$. Then the distance of the point $\mathrm { P } ( \mathrm { m } , \mathrm { n } )$ from the point $\mathrm { Q } ( - 2 , - 3 )$ is
(1) 10
(2) 6
(3) 4
(4) 8

$$\mathrm { X } \subseteq \{ \mathrm { a } , \mathrm {~b} , \mathrm { c } , \mathrm {~d} , \mathrm { e } \}$$
Given that, how many different subsets $X$ are there such that the number of elements in $\mathbf { X } \cap \{ \mathbf { a } , \mathbf { b } \}$ is 1?
A) 10 B) 12 C) 14 D) 16 E) 18

Let A be a subset of the set $\{ 1,2,3,4,5,6,7 \}$. $$A \cap \{ 5,6,7 \}$$ The elements of the set are odd numbers.\ Accordingly, how many three-element sets A satisfy this condition?\ A) 12\ B) 14\ C) 16\ D) 18\ E) 20

Let A and B be non-empty sets consisting of digits. If
$$A \cap B = A \cap \{ 0,2,4,6,8 \}$$
equality is satisfied, then A is called the common-intersection set of B. Given that set A is the common-intersection set of
$$B = \{ 0,1,2,3,4 \}$$
how many different sets A are there?
A) 3
B) 7
C) 15
D) 31
E) 63