Let $S = \{ 1,2,3,4,5,6,9 \}$. Then the number of elements in the set $T = \{ A \subseteq S : A \neq \phi$ and the sum of all the elements of $A$ is not a multiple of $3 \}$ is

The number of bijective functions $f:\{1,3,5,7,\cdots,99\} \rightarrow \{2,4,6,8,\cdots,100\}$ if $f(3) > f(5) > f(7) \cdots > f(99)$ is
(1) ${}^{50}C_1$
(2) ${}^{50}C_2$
(3) $\frac{50!}{2}$
(4) ${}^{50}C_3 \times 3!$

There are ten boys $B _ { 1 } , B _ { 2 } , \ldots , B _ { 10 }$ and five girls $G _ { 1 } , G _ { 2 } , \ldots G _ { 5 }$ in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both $B _ { 1 }$ and $B _ { 2 }$ together should not be the members of a group, is $\_\_\_\_$.

A class contains $b$ boys and $g$ girls. If the number of ways of selecting 3 boys and 2 girls from the class is 168, then $b + 3g$ is equal to $\_\_\_\_$.

Eight persons are to be transported from city $A$ to city $B$ in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is
(1) 1120
(2) 3360
(3) 1680
(4) 560

The number of ways of selecting two numbers $a$ and $b$, $a \in \{2, 4, 6, \ldots, 100\}$ and $b \in \{1, 3, 5, \ldots, 99\}$ such that 2 is the remainder when $a + b$ is divided by 23 is
(1) 186
(2) 54
(3) 108
(4) 268

Let $x$ and $y$ be distinct integers where $1 \leq x \leq 25$ and $1 \leq y \leq 25$. Then, the number of ways of choosing $x$ and $y$, such that $x + y$ is divisible by 5 , is $\_\_\_\_$ .

If all the six digit numbers $\mathrm { x } _ { 1 } \mathrm { x } _ { 2 } \mathrm { x } _ { 3 } \mathrm { x } _ { 4 } \mathrm { x } _ { 5 } \mathrm { x } _ { 6 }$ with $0 < \mathrm { x } _ { 1 } < \mathrm { x } _ { 2 } < \mathrm { x } _ { 3 } < \mathrm { x } _ { 4 } < \mathrm { x } _ { 5 } < \mathrm { x } _ { 6 }$ are arranged in the increasing order, then the sum of the digits in the $72 ^ { \text {th} }$ number is $\_\_\_\_$ .

Let $S = \{ 1,2,3,5,7,10,11 \}$. The number of non-empty subsets of $S$ that have the sum of all elements a multiple of 3 , is $\_\_\_\_$ .

Suppose Anil's mother wants to give 5 whole fruits to Anil from a basket of 7 red apples, 5 white apples and 8 oranges. If in the selected 5 fruits, at least 2 orange, at least one red apple and at least one white apple must be given, then the number of ways, Anil's mother can offer 5 fruits to Anil is $\_\_\_\_$.

The number of ways, in which 5 girls and 7 boys can be seated at a round table so that no two girls sit together is
(1) 720
(2) $126 ( 5 ! ) ^ { 2 }$
(3) $7 ( 360 ) ^ { 2 }$
(4) $7 ( 720 ) ^ { 2 }$

The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48 , is (1) 472 (2) 432 (3) 507 (4) 400

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

The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is
(1) 48
(2) 56
(3) 24
(4) 16

The number of ways five alphabets can be chosen from the alphabets of the word MATHEMATICS, where the chosen alphabets are not necessarily distinct, is equal to : (1) 179 (2) 177 (3) 181 (4) 175

The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is
(1) 406
(2) 130
(3) 142
(4) 136

Let the set $S = \{ 2,4,8,16 , \ldots , 512 \}$ be partitioned into 3 sets $A , B , C$ with equal number of elements such that $\mathrm { A } \cup \mathrm { B } \cup \mathrm { C } = \mathrm { S }$ and $\mathrm { A } \cap \mathrm { B } = \mathrm { B } \cap \mathrm { C } = \mathrm { A } \cap \mathrm { C } = \phi$. The maximum number of such possible partitions of $S$ is equal to:
(1) 1680
(2) 1640
(3) 1520
(4) 1710

The lines $L_1, L_2, \ldots, L_{20}$ are distinct. For $n = 1, 2, 3, \ldots, 10$ all the lines $L_{2n-1}$ are parallel to each other and all the lines $L_{2n}$ pass through a given point $P$. The maximum number of points of intersection of pairs of lines from the set $\{L_1, L_2, \ldots, L_{20}\}$ is equal to:

There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is $\_\_\_\_$

The number of integers, between 100 and 1000 having the sum of their digits equals to 14, is $\_\_\_\_$

In an examination of Mathematics paper, there are 20 questions of equal marks and the question paper is divided into three sections: A, B and C. A student is required to attempt total 15 questions taking at least 4 questions from each section. If section A has 8 questions, section B has 6 questions and section C has 6 questions, then the total number of ways a student can select 15 questions is $\underline{\hspace{1cm}}$.

Group A consists of 7 boys and 3 girls, while group $B$ consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group $A$ and the remaining 3 from group $B$, is equal to :
(1) 8750
(2) 9100
(3) 8925
(4) 8575

Let ${ } ^ { \mathrm { n } } \mathrm { C } _ { \mathrm { r } - 1 } = 28 , { } ^ { \mathrm { n } } \mathrm { C } _ { \mathrm { r } } = 56$ and ${ } ^ { \mathrm { n } } \mathrm { C } _ { \mathrm { r } + 1 } = 70$. Let $\mathrm { A } ( 4 \cos t , 4 \sin t ) , \mathrm { B } ( 2 \sin t , - 2 \cos \mathrm { t } )$ and $C \left( 3 r - n , r ^ { 2 } - n - 1 \right)$ be the vertices of a triangle $ABC$, where $t$ is a parameter. If $( 3 x - 1 ) ^ { 2 } + ( 3 y ) ^ { 2 } = \alpha$, is the locus of the centroid of triangle ABC, then $\alpha$ equals
(1) 6
(2) 18
(3) 8
(4) 20

Consider the region $R = \left\{(x, y) : x \leq y \leq 9 - \frac{11}{3}x^2,\, x \geq 0\right\}$. The area of the largest rectangle of sides parallel to the coordinate axes and inscribed in $R$, is:
(1) $\frac{730}{119}$
(2) $\frac{625}{111}$
(3) $\frac{821}{123}$
(4) $\frac{567}{121}$

Q62. The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is
(1) 48
(2) 56
(3) 24
(4) 16