How many functions $f : \{ 1,2 , \ldots , 10 \} \rightarrow \{ 1 , \ldots , 2000 \}$, which satisfy $$f ( i + 1 ) - f ( i ) \geq 20 , \text { for all } 1 \leq i \leq 9 ,$$ are there?
(A) $10 ! \binom { 1829 } { 10 }$
(B) $11 ! \binom { 1830 } { 11 }$
(C) $\binom { 1829 } { 10 }$
(D) $\binom { 1830 } { 11 }$

Suppose 40 distinguishable balls are to be distributed into 4 different boxes such that each box gets exactly 10 balls. Out of these 40 balls, 10 are defective and 30 are non-defective. In how many ways can the balls be distributed such that all the defective balls go to the first two boxes?
(A) $\frac{40!}{(10!)^4}$
(B) $\frac{30! \cdot 20!}{(10!)^5}$
(C) $\frac{20! \cdot 20!}{(10!)^5}$
(D) $\frac{30! \cdot 10!}{(10!)^4}$

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) $2 ^ { 9 }$

A box contains 10 red cards numbered $1 , \ldots , 10$ and 10 black cards numbered $1 , \ldots , 10$. In how many ways can we choose 10 out of the 20 cards so that there are exactly 3 matches, where a match means a red card and a black card with the same number?
(a) $\binom { 10 } { 3 } \binom { 7 } { 4 } 2 ^ { 4 }$.
(B) $\binom { 10 } { 3 } \binom { 7 } { 4 }$.
(C) $\binom { 10 } { 3 } 2 ^ { 7 }$.
(D) $\binom { 10 } { 3 } \binom { 14 } { 4 }$.

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 $n \geq 2$ be an integer. Take $n$ distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining every pair of adjacent points by blue and the rest by red. If the number of red and blue line segments are equal, then the value of $n$ is

Let $n_1 < n_2 < n_3 < n_4 < n_5$ be positive integers such that $n_1 + n_2 + n_3 + n_4 + n_5 = 20$. Then the number of such distinct arrangements $(n_1, n_2, n_3, n_4, n_5)$ is

A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is
(A) 380
(B) 320
(C) 260
(D) 95

Let $S = \{ 1,2,3 , \ldots , 9 \}$. For $k = 1,2 , \ldots , 5$, let $N _ { k }$ be the number of subsets of $S$, each containing five elements out of which exactly $k$ are odd. Then $N _ { 1 } + N _ { 2 } + N _ { 3 } + N _ { 4 } + N _ { 5 } =$
[A] 210
[B] 252
[C] 125
[D] 126

Let $X$ be a set with exactly 5 elements and $Y$ be a set with exactly 7 elements. If $\alpha$ is the number of one-one functions from $X$ to $Y$ and $\beta$ is the number of onto functions from $Y$ to $X$, then the value of $\frac { 1 } { 5 ! } ( \beta - \alpha )$ is $\_\_\_\_$ .

In a high school, a committee has to be formed from a group of 6 boys $M _ { 1 } , M _ { 2 } , M _ { 3 } , M _ { 4 } , M _ { 5 } , M _ { 6 }$ and 5 girls $G _ { 1 } , G _ { 2 } , G _ { 3 } , G _ { 4 } , G _ { 5 }$.
(i) Let $\alpha _ { 1 }$ be the total number of ways in which the committee can be formed such that the committee has 5 members, having exactly 3 boys and 2 girls.
(ii) Let $\alpha _ { 2 }$ be the total number of ways in which the committee can be formed such that the committee has at least 2 members, and having an equal number of boys and girls.
(iii) Let $\alpha _ { 3 }$ be the total number of ways in which the committee can be formed such that the committee has 5 members, at least 2 of them being girls.
(iv) Let $\alpha _ { 4 }$ be the total number of ways in which the committee can be formed such that the committee has 4 members, having at least 2 girls and such that both $M _ { 1 }$ and $G _ { 1 }$ are NOT in the committee together.
LIST-I P. The value of $\alpha _ { 1 }$ is Q. The value of $\alpha _ { 2 }$ is R. The value of $\alpha _ { 3 }$ is S. The value of $\alpha _ { 4 }$ is
LIST-II

1. 136
2. 189
3. 192
4. 200
5. 381
6. 461

The correct option is:
(A) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 6 ; ~ } \mathbf { R } \rightarrow \mathbf { 2 ; } \mathbf { S } \rightarrow \mathbf { 1 }$
(B) $\mathbf { P } \rightarrow \mathbf { 1 } ; \mathbf { Q } \rightarrow \mathbf { 4 } ; \mathbf { R } \rightarrow \mathbf { 2 } ; \mathbf { S } \rightarrow \mathbf { 3 }$
(C) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 6 } ; \mathbf { R } \rightarrow \mathbf { 5 } ; \mathbf { S } \rightarrow \mathbf { 2 }$
(D) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 2 } ; \mathbf { R } \rightarrow \mathbf { 3 } ; \mathbf { S } \rightarrow \mathbf { 1 }$

An engineer is required to visit a factory for exactly four days during the first 15 days of every month and it is mandatory that no two visits take place on consecutive days. Then the number of all possible ways in which such visits to the factory can be made by the engineer during 1-15 June 2021 is $\_\_\_\_$

Let $$\begin{gathered} S _ { 1 } = \{ ( i , j , k ) : i , j , k \in \{ 1,2 , \ldots , 10 \} \} , \\ S _ { 2 } = \{ ( i , j ) : 1 \leq i < j + 2 \leq 10 , i , j \in \{ 1,2 , \ldots , 10 \} \} , \\ S _ { 3 } = \{ ( i , j , k , l ) : 1 \leq i < j < k < l , \quad i , j , k , l \in \{ 1,2 , \ldots , 10 \} \} \end{gathered}$$ and $$S _ { 4 } = \{ ( i , j , k , l ) : i , j , k \text { and } l \text { are distinct elements in } \{ 1,2 , \ldots , 10 \} \} .$$ If the total number of elements in the set $S _ { r }$ is $n _ { r } , r = 1,2,3,4$, then which of the following statements is (are) TRUE ?
(A) $n _ { 1 } = 1000$
(B) $n _ { 2 } = 44$
(C) $n _ { 3 } = 220$
(D) $\frac { n _ { 4 } } { 12 } = 420$

Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from these 4 boxes so that from each box at least one red ball and one blue ball are chosen ?
(A) 21816
(B) 85536
(C) 12096
(D) 156816

A group of 9 students, $s _ { 1 } , s _ { 2 } , \ldots , s _ { 9 }$, is to be divided to form three teams $X , Y$, and $Z$ of sizes 2,3 , and 4 , respectively. Suppose that $s _ { 1 }$ cannot be selected for the team $X$, and $s _ { 2 }$ cannot be selected for the team $Y$. Then the number of ways to form such teams, 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 $n ( X ) = {}^{ m } C _ { 6 }$, 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 $\_\_\_\_$ .

The set $S = \{ 1,2,3 , \ldots , 12 )$ is to be partitioned into three sets $A , B , C$ of equal size. Thus, $A \cup B \cup C = S , A \cap B = B \cap C = A \cap C = \phi$. The number of ways to partition $S$ is
(1) $\frac { 12 ! } { 3 ! ( 4 ! ) ^ { 3 } }$
(2) $\frac { 12 ! } { 3 ! ( 3 ! ) ^ { 4 } }$
(3) $\frac { 12 ! } { ( 4 ! ) ^ { 3 } }$
(4) $\frac { 12 ! } { ( 3 ! ) ^ { 4 } }$

This question has Statement-1 and Statement-2. Of the four choices given after the statements, choose the one that best describes the two statements. Statement-1: The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is ${}^{9}\mathrm{C}_{3}$. Statement-2: The number of ways of choosing any 3 places from 9 different places is ${}^{9}\mathrm{C}_{3}$.
(1) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(2) Statement-1 is true, Statement-2 is false.
(3) Statement-1 is false, Statement-2 is true.
(4) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

If the number of 5-element subsets of the set $A = \left\{ a _ { 1 } , a _ { 2 } , \ldots , a _ { 20 } \right\}$ of 20 distinct elements is $k$ times the number of 5-element subsets containing $a _ { 4 }$, then $k$ is
(1) 5
(2) $\frac { 20 } { 7 }$
(3) 4
(4) $\frac { 10 } { 3 }$

If $n = { } ^ { m } C _ { 2 }$, then the value of ${ } ^ { n } C _ { 2 }$ is given by
(1) $3 \left( { } ^ { m + 1 } C _ { 4 } \right)$
(2) ${ } ^ { m - 1 } C _ { 4 }$
(3) ${ } ^ { m + 1 } C _ { 4 }$
(4) $2 \left( { } ^ { m + 2 } C _ { 4 } \right)$

Statement 1: If $A$ and $B$ be two sets having $p$ and $q$ elements respectively, where $q > p$. Then the total number of functions from set $A$ to set $B$ is $q^{p}$. Statement 2: The total number of selections of $p$ different objects out of $q$ objects is ${}^{q}C_{p}$.
(1) Statement 1 is true, Statement 2 is false.
(2) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.
(3) Statement 1 is false, Statement 2 is true.
(4) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation of Statement 1.

A committee of 4 persons is to be formed from 2 ladies, 2 old men and 4 young men such that it includes at least 1 lady, at least 1 old man and at most 2 young men. Then the total number of ways in which this committee can be formed is :
(1) 40
(2) 41
(3) 16
(4) 32

Let $T_n$ be the number of all possible triangles formed by joining vertices of an $n$-sided regular polygon. If $T_{n+1} - T_n = 10$, then the value of $n$ is:
(1) 10
(2) 8
(3) 5
(4) 7

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