Let $A = \{2, 3, 4\}$ and $B = \{8, 9, 12\}$. Then the number of elements in the relation $R = \{ ( ( a _ { 1 } , b _ { 1 } ) , ( a _ { 2 } , b _ { 2 } ) ) \in ( A \times B ) \times ( A \times B ) : a _ { 1 }$ divides $b _ { 2 }$ and $a _ { 2 }$ divides $b _ { 1 } \}$ is
(1) 36
(2) 24
(3) 18
(4) 12

The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is $\_\_\_\_$.

Let 5 digit numbers be constructed using the digits $0, 2, 3, 4, 7, 9$ with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number 42923 is $\underline{\hspace{1cm}}$.

The number of permutations, of the digits $1, 2, 3, \ldots, 7$ without repetition, which neither contain the string 153 nor the string 2467, is $\_\_\_\_$.

Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11, is equal to $\underline{\hspace{1cm}}$.

Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to
(1) 18
(2) 16
(3) 12
(4) 15

60 words can be made using all the letters of the word BHBJO, with or without meaning. If these words are written as in a dictionary, then the $50 ^ { \text {th} }$ word is :
(1) JBBOH
(2) OBBJH
(3) OBBHJ
(4) HBBJO

If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at $315 ^ { \text {th} }$ position in this arrangement is:
(1) NRAGUP
(2) NRAPUG
(3) NRAPGU
(4) NRAGPU

Let $[ t ]$ be the greatest integer less than or equal to $t$. Let $A$ be the set of all prime factors of 2310 and $f : A \rightarrow \mathbb { Z }$ be the function $f ( x ) = \left[ \log _ { 2 } \left( x ^ { 2 } + \left[ \frac { x ^ { 3 } } { 5 } \right] \right) \right]$. The number of one-to-one functions from $A$ to the range of $f$ is
(1) 25
(2) 24
(3) 20
(4) 120

The number of 3-digit numbers, formed using the digits $2,3,4,5$ and 7, when the repetition of digits is not allowed, and which are not divisible by 3, is equal to $\_\_\_\_$

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

All the letters of the word $G T W E N T Y$ are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word $G T W E N T Y$ IS

In a group of 3 girls and 4 boys, there are two boys $B _ { 1 }$ and $B _ { 2 }$. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $B _ { 1 }$ and $B _ { 2 }$ are not adjacent to each other, is :
(1) 96
(2) 144
(3) 120
(4) 72

Let S be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set S, one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is :
(1) $\frac { 1 } { 2 }$
(2) $\frac { 1 } { 4 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { 1 } { 3 }$

Let P be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in P are formed by using the digits 1, 2 and 3 only, then the number of elements in the set $P$ is:
(1) 173
(2) 164
(3) 158
(4) 161

If all the words with or without meaning made using all the letters of the word ``KANPUR'' are arranged as in a dictionary, then the word at $440 ^ { \text {th} }$ position in this arrangement, is :
(1) PRNAUK
(2) PRKANU
(3) PRKAUN
(4) PRNAKU

The number of words, which can be formed using all the letters of the word ``DAUGHTER'', so that all the vowels never come together, is
(1) 36000
(2) 37000
(3) 34000
(4) 35000

The number of different 5 digit numbers greater than 50000 that can be formed using the digits $0, 1, 2, 3, 4, 5, 6, 7$, such that the sum of their first and last digits should not be more than 8, is
(1) 4608
(2) 5720
(3) 5719
(4) 4607

The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is

The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is

The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is \_\_\_\_ .

Number of functions $f : \{1, 2, \ldots, 100\} \rightarrow \{0, 1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to $\_\_\_\_$.

Q1 Using the five numerals $0,1,2,3,4$, we are to make four-digit integers. (Note that "0123", etc. are not allowed.)
(1) The total possible number of integers where the digits are all different numerals is $\mathbf{AB}$. Among them, the total number of integers that do not use 0 is $\mathbf{CD}$.
(2) If we are allowed to use the same numeral repeatedly, then the total possible number of four-digit integers is $\mathbf{EFG}$. Among them
(i) the total number of integers that use both 1 and 3 twice is $\mathbf{H}$,
(ii) the total number of integers that use both 0 and 4 twice is $\mathbf{I}$,
(iii) the total number of integers that use both of two numerals twice is $\mathbf{JK}$.

Consider the permutations of the eight letters of the word ``POSITION''.
(1) The number of permutations in which the two I's are adjacent and the two O's also are adjacent is $\mathbf { A B C }$.
(2) The number of permutations such that the permutations both begin and end with the letter I and furthermore the two O's are adjacent is $\mathbf{DEF}$.
(3) The number of permutations that both begin and end with the letter I is $\mathbf{GHI}$.
(4) The number of permutations of the 4 letters I, I, O, O is $\mathbf { J }$. Also, the number of permutations of the 4 letters $\mathrm { N } , \mathrm { P } , \mathrm { S } , \mathrm { T }$ is $\mathbf { K L }$.
Hence the number of permutations of POSITION which begin or end with either I or O, and furthermore in which none of letters $\mathrm { N } , \mathrm { P } , \mathrm { S } , \mathrm { T }$ are adjacent to each other is $\mathbf{MNO}$.

We have four white cards, three red cards and three black cards. A different number is written on each of the ten cards.
(1) Choose two of the ten cards and put one in box A, and one in box B. There are $\mathbf{NO}$ ways of putting two cards in the two boxes.
(2) There are $\mathbf { P Q }$ ways of choosing two cards of the same color, and $\mathbf { R S }$ ways of choosing two cards of different colors.
Next, put the ten cards in a box and take out one card and without returning it to the box, take out second card.
(3) The probability that the two cards taken out have the same color is $\square\mathbf{ T UV}$
(4) The probability that the color of the first card taken out is white or red, and the color of the second card taken out is red or black is $\frac { \mathbf { W X } } { \mathbf { Y } \mathbf { Y } }$.