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

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

Let $A = \{ 1,2,3 , \ldots , 10 \}$ and $B = \left\{ \frac { m } { n } : m , n \in A , m < n \right.$ and $\left. \operatorname { gcd } ( m , n ) = 1 \right\}$. Then $n ( B )$ is equal to:
(1) 36
(2) 31
(3) 37
(4) 29

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

Let $S = \{p_1, p_2, \ldots, p_{10}\}$ be the set of first ten prime numbers. Let $A = S \cup P$, where $P$ is the set of all possible products of distinct elements of $S$. Then the number of all ordered pairs $(x, y)$, $x \in S$, $y \in A$, such that $x$ divides $y$, is $\underline{\hspace{2cm}}$.

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 $\_\_\_\_$.

The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is $\underline{\hspace{2cm}}$.

Q62. 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

Q63. 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

Q69. 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

Q81. The number of ways of getting a sum 16 on throwing a dice four times is $\_\_\_\_$

Q81. 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 $\_\_\_\_$

Q81. The number of integers, between 100 and 1000 having the sum of their digits equals to 14 , is $\_\_\_\_$

Q87. 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 $\_\_\_\_$

If words are arranged in a dictionary alphabetically, then rank of UDAYPUR is?

Number of 4 letters words with or without meaning formed from the letters of the word PQRSSSTTUVV is\ (A) 1232\ (B) 1422\ (C) 1400\ (D) 1162

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 $\mathbf{JK}$.

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.

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 $\frac { \mathbf { W X } } { \mathbf { Y } \mathbf { Y } }$.

We have four cards of different sizes. We are to paint each card either red, black, blue or yellow. However, we may paint more than one card with the same color.
(1) There are a total of $\mathbf { N O P }$ ways of painting the cards.
(2) There are $\mathbf{QR}$ ways of painting them using all four colors.
(3) There are $\mathbf{ST}$ ways of painting two cards red, one card black, and one card blue.
(4) There are $\mathbf{UVW}$ ways of painting the four cards using three colors.
(5) There are $\mathbf { X Y }$ ways of painting them using two colors.

We have four cards of different sizes. We are to paint each card either red, black, blue or yellow. However, we may paint more than one card with the same color.
(1) There are a total of $\mathbf { N O P }$ ways of painting the cards.
(2) There are $\mathbf{QR}$ ways of painting them using all four colors.
(3) There are $\mathbf{ST}$ ways of painting two cards red, one card black, and one card blue.
(4) There are $\mathbf{UVW}$ ways of painting the four cards using three colors.
(5) There are $\mathbf{XY}$ ways of painting them using two colors.