If $\sum_{k=1}^{31} \left({}^{31}\mathrm{C}_k\right)\left({}^{31}\mathrm{C}_{k-1}\right) - \sum_{k=1}^{30}\left({}^{30}\mathrm{C}_k\right)\left({}^{30}\mathrm{C}_{k-1}\right) = \frac{\alpha(60!)}{(30!)(31!)}$, where $\alpha \in R$, then the value of $16\alpha$ is equal to
(1) 1411
(2) 1320
(3) 1615
(4) 1855

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

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 $\_\_\_\_$ .

The number of 4-letter words, with or without meaning, each consisting of 2 vowels and 2 consonants, which can be formed from the letters of the word UNIVERSE without repetition 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 $\_\_\_\_$.

Five digit numbers are formed using the digits $1,2,3,5,7$ with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1. Then the serial number of 35337 is

$\sum _ { k = 0 } ^ { 6 } { } ^ { 51 - k } C _ { 3 }$ is equal to
(1) ${ } ^ { 51 } C _ { 4 } - { } ^ { 45 } C _ { 4 }$
(2) ${ } ^ { 51 } C _ { 3 } - { } ^ { 45 } C _ { 3 }$
(3) ${ } ^ { 52 } C _ { 4 } - { } ^ { 45 } C _ { 4 }$
(4) ${ } ^ { 52 } C _ { 3 } - { } ^ { 45 } C _ { 3 }$

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

If for some $m, n$; ${}^{6}C_m + 2\,{}^{6}C_{m+1} + {}^{6}C_{m+2} > {}^{8}C_3$ and ${}^{n-1}P_3 : {}^{n}P_4 = 1 : 8$, then ${}^{n}P_{m+1} + {}^{n+1}C_m$ is equal to
(1) 380
(2) 376
(3) 384
(4) 372

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

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 $\_\_\_\_$

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 $\underline{\hspace{1cm}}$.

Let $A = \{ ( x , y ) : 2 x + 3 y = 23 , x , y \in \mathbb { N } \}$ and $B = \{ x : ( x , y ) \in A \}$. Then the number of one-one functions from $A$ to $B$ is equal to $\_\_\_\_$

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

From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is:
(1) 5148
(2) 6084
(3) 4356
(4) 14950

There are six boxes numbered from 1 to 6. We are to put four balls of different sizes into these boxes.
(1) There are altogether $\mathbf{AA}$ ways to put the four balls into the boxes.
(2) There are $\mathbf{CDE}$ ways to put the four balls into four separate boxes.
(3) There are $\mathbf{FGH}$ ways to put three balls into one box and the fourth ball into another.
(4) There are $\mathbf{IJK}$ ways to put at least one ball into the box numbered 1.