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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 ________ .

Let \( A=\{1,3,7,9,11\} \) and \( B=\{2,4,5,7,8,10,12\} \). Then the total number of one-one maps \( f:A\to B \), such that \( f(1)+f(3)=14 \), is:

A box contains \( 5 \) blue, \( 6 \) yellow and \( 4 \) red balls. The number of ways, of drawing \( 8 \) balls containing at least two balls of each colour, is:

Number of ways of arranging \( 8 \) identical books into \( 4 \) identical shelves where any number of shelves may remain empty is equal to:

The number of elements in the set \( S=\{(x,y,z):x,y,z\in\mathbb{Z},\;x+2y+3z=42,\;x,y,z\ge 0\} \) equals ______.

The sum of all the four-digit numbers that can be formed using all the digits \( 2 \), \( 1 \), \( 2 \), \( 3 \) is equal to ______.

Let \( S=\{1,2,3,4,5,6,7\} \). Then the number of possible functions \( f:S\to S \) such that \( f(m\cdot n)=f(m)\cdot f(n) \) for every \( m,n\in S \) and \( m\cdot n\in S \) is equal to \( \underline{\quad} \).

The number of ways, in which the letters \( A,B,C,D,E \) can be placed in the \( 8 \) boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is:

Let \( S = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \). Let \( x \) be the number of \( 9 \)-digit numbers formed using the digits of the set \( S \) such that only one digit is repeated and it is repeated exactly twice. Let \( y \) be the number of \( 9 \)-digit numbers formed using the digits of the set \( S \) such that only two digits are repeated and each of these is repeated exactly twice. Then,

All the arrangements, with or without meaning, of the word FARMER are written excluding any word that has two R appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word FARMER in this list is:

The number of functions \( f : \{1,2,3,4\} \to \{a,b,c\} \), which are not onto, is:

The number of ways, of forming a queue of \( 4 \) boys and \( 3 \) girls such that all the girls are not together, is:

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 ____________.

The number of five digit numbers, greater than \( 40000 \) and divisible by \( 5 \), which can be formed using the digits \( 0, 1, 3, 5, 7 \) and \( 9 \) without repetition, is equal to:

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

There are \( 4 \) men \( 5 \) 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 ways, \( 16 \) identical cubes, of which \( 11 \) are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least \( 2 \) blue cubes, 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 seven digits odd numbers, that can be formed using all the seven digits \( 1, 2, 2,2, 3, 3, 5 \) is ________.

The number of \( 7 \)-digit numbers which are multiples of \( 11 \) and are formed using all the digits \( 1,2,3,4,5,7 \) and \( 9 \) is ________.

The number of strictly increasing functions \( f \) from the set \( \{1, 2, 3, 4, 5, 6\} \) to the set \( \{1, 2, 3, \ldots, 9\} \) such that \( f(i) \neq i \) for \( 1 \le i \le 6 \), is equal to:

The number of \( 3 \)-digit odd numbers, whose sum of digits is a multiple of \( 7 \), is ___________.

Let \( p_n \) denote the total number of triangles formed by joining the vertices of an \( n \)-side regular polygon. If \( p_{n+1} - p_n = 66 \), then the sum of all distinct prime divisors of \( n \) is:

The number is integers, between \( 100 \) and \( 1000 \) having the sum of their digits equals to \( 14 \), is ________.

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:

Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is \( 840 \), then the total numbers of persons, who participated in the tournament, is ________.

The number of one-one functions \( f : \{a, b, c, d\} \to \{0, 1, 2, \ldots, 10\} \) such that \( 2f(a) - f(b) + 3f(c) + f(d) = 0 \) is ________.

The number of \( 4 \)-letter words, with or without meaning, which can be formed using the letters PQRPQRSTUVP, is ______.

Let \( n \) a non-negative integer. Then the number of divisors of the form “\( 4n+1 \)” of the number \( 10^{(10)}\cdot (11)^{11}\cdot (13)^{13} \) is equal to ____________.

The number of \( 3 \)-digit numbers, that are divisible by \( 2 \) and \( 3 \), but not divisible by \( 4 \) and \( 9 \), is _______.

The number of \( 4 \)-letter words, with or without meaning, each consisting of two vowels and two consonants that can be formed the letters of the word INCONSEQUENTIAL, without repeating any letter, is:

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:

Let \( m \) and \( n \), \( (m < n\)) be two 2-digit numbers.Then the total numbers of pairs (m,n) such that gcd (m,n) = 6, is ______________.

Line \( L_1 \) of slope \( 2 \) and \( L_2 \) of slope \( \frac{1}{2} \) intersect at the origin \( O \). In the first quadrant, \( P_1, P_2, \ldots, P_{12} \) are \( 12 \) points on line \( L_1 \) and \( Q_1, Q_2, \ldots, Q_9 \) are \( 9 \) points on line \( L_2 \). Then the total number of triangles, that can be formed having vertices at three of the \( 22 \) points \( O, P_1, P_2, \ldots, P_{12}, Q_1, Q_2, \ldots, Q_9 \), is:

The largest \( n \in \mathbb{N} \) such that \( 3^{n} \) divides \( 50! \) is:

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:

The total number of three-digit numbers, divisible by \( 3 \), which can be formed using the digits \( 1,3,5,8 \), if repetition of digits is allowed, is:

There are \( 12 \) points in a plane, no three of which are in the same straight line, except \( 5 \) points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these \( 12 \) points is:

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 __________ .

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 ______.

All words, with or without meaning, are made using all the letters of the word MONDAY. These words are written as in a dictionary with serial numbers. The serial number of the word MONDAY is :

The number of six letter words (with or without meaning), formed using all the letters of the word ‘VOWELS’, so that all the consonants never come together, is __________ .

The number of numbers, strictly between \( 5000 \) and \( 10000 \) can be formed using the digit \( 1,3,5,7,9 \) without repetition, is:

Let \( b_1 b_2 b_3 b_4 \) be a \( 4 \)-element permutation with \( b_i \in \{1, 2, 3, \ldots, 100\} \) for \( 1 \le i \le 4 \) and \( b_i \ne b_j \) for \( i \ne j \), such that either \( b_1, b_2, b_3 \) are consecutive integers or \( b_2, b_3, b_4 \) are consecutive integers. Then the number of such permutations \( b_1 b_2 b_3 b_4 \) is equal to ________.

Let \( P \) be the set of seven digit numbers with sum of their digit 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:

Consider a rectangle \( ABCD \) having \( 5,\,7,\,6,\,9 \) points in the interior of the line segments \( AB,\, CD,\, BC,\, DA \) respectively. Let \( \alpha \) be the number of triangles having these points from different sides as vertices and \( \beta \) be the number of quadrilaterals having these points from different sides as vertices. Then \(( \beta - \alpha )\) is equal to:

In an examination, \( 5 \) students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is ______.

A building has ground floor and \( 10 \) more floors. Nine persons enter in a lift at the ground floor. The lift goes up to the \( 10^{\text{th}} \) floor. The number of ways, in which any \( 4 \) persons exit at a floor and the remaining \( 5 \) persons exit at a different floor, if the lift does not stop at the first and the second floors, is equal to:

Let \( ABC \) be a triangle. Consider four points \( p_1, p_2, p_3, p_4 \) on the side \( AB \), five points \( p_5, p_6, p_7, p_8, p_9 \) on the side \( BC \), and four points \( p_{10}, p_{11}, p_{12}, p_{13} \) on the side \( AC \). None of these points is a vertex of the triangle \( ABC \). Then the total number of pentagons, that can be formed by taking all the vertices from the points \( p_1, p_2, \ldots, p_{13} \), is __________.

The number of integers, greater than \( 7000 \) that can be formed, using the digits \( 3, 5, 6, 7, 8 \) without repetition, is:

The total number of \( 4 \)-digit numbers whose greatest common divisor with \( 54 \) is \( 2 \), is ______.

The letters of the word “UDAYPUR” are written in all possible ways with or without meaning and these words are arranged as in a dictionary. The rank of the word “UDAYPUR” is:

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:

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 ______.

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:

Let \( P_{1}, P_{2}, \ldots , P_{15} \) be \( 15 \) points on a circle. The number of distinct triangles formed by points \( P_{i}, P_{j}, P_{k} \) such that \( i + j + k \neq 15 \), is:

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:

The number of ways to distribute \(30\) identical candies among four children \(C_1, C_2, C_3\) and \(C_4\) so that \(C_2\) receives atleast \(4\) and atmost \(7\) candies, \(C_3\) receives atleast \(2\) and atmost \(6\) candies, is equal to:

Let \( \alpha=\dfrac{(4!)!}{(4!)^{3!}} \) and \( \beta=\dfrac{(5!)!}{(5!)^{4!}} \). Then:

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:

If \( n \) is the number of ways five different employees can sit into four indistinguished offices where any office may have any number of persons including zero, then \( n \) is equal to:

If the number of words, with or without meaning, which can be made using all the letters of the word MATHEMATICS in which C and S do not come together, is \( (6!){k} \) then \( k \) is equal to:

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 ________.

The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is:

There are \( 15 \) players in a cricket team, out of which \( 6 \) are bowlers, \( 7 \) are batsmen and \( 2 \) are wicketkeepers. The number of ways, a team of \( 11 \) players be selected from them so as to include at least \( 4 \) bowlers, \( 5 \) batsmen and \( 1 \) wicketkeeper, is _______.

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:

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 \( 2 \) oranges, 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 ____.

Number of functions \( f : \{1,2,\ldots,100\} \to \{0,1\} \), that assign \( 1 \) to exactly one of the positive integers less than or equal to \( 98 \), is equal to \( \underline{\phantom{xxxx}} \).

A person forgets his \( 4 \)-digit ATM pin code. But he remembers that in the code all the digits are different, the greatest digit is \( 7 \) and the sum of the first two digits is equal to the sum of the last two digits. Then the maximum number of trials necessary to obtain the correct code is \( \_\_\_\_\_\_\_\_ \).

A number is called a palindrome if it reads the same backward as well as forward. For example \( 285582 \) is a six digit palindrome. The number of six digit palindromes, which are divisible by \( 55 \), is ________.

If the number of seven-digit numbers, such that the sum of their digits is even, is \( m \cdot n \cdot 10^{n} \); \( m,n \in \{1,2,3,\ldots,9\} \), then \( m+n \) is equal to ________.

The number of ways, in which \( 16 \) oranges can be distributed to four children such that each child gets at least one orange, is:

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 \( A\cup B\cup C=S \) and \( A\cap B=B\cap C=A\cap C=\phi. \). The maximum number of such possible partitions of \( S \) is equal to:

The sum of all the \( 4 \)-digit distinct numbers that can be formed with the digits \( 1 \), \( 2 \), \( 2 \) and \( 3 \) is:

Total number of \( 3 \)-digit numbers that are divisible by \( 6 \) and can be formed by using the digits \( 1 \), \( 2 \), \( 3 \), \( 4 \), \( 5 \) with repetition, is _________.

The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is:

The number of natural numbers lying between \( 1012 \) and \( 23421 \) that can be formed using the digits \( 2, 3, 4, 5, 6 \) (repetition of digits is not allowed) and divisible by \( 55 \) is ________.

Let \( S=\{(m,n): m,n\in \{1,2,3,\ldots,50\}\} \). If the number of elements \( (m,n) \) is \( S \) such that \( 6^{m}+9^{n} \) is a multiple of \( 5 \) is \( p \) and the number of elements \( (m,n) \) in \( S \) such that \( m+n \) is a square of a prime number is \( q \), then \( p+q \) is equal to _________.

If $^{2n+1}P_{n-1} : {^{2n-1}P_n} = 11 : 21$, then $n^2 + n + 15$ is equal to ________.

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 students 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 \( C \) \( 6 \) questions, then the total number of ways a student can select \( 15 \) questions is ______.

If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial numbers, then the serial number of the word THAMS 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 ______________.

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 ________.

The number of \( 9 \)-digit numbers, that can be formed using all the digits of the number \( 123412341 \) so that the even digits occupy only even places, is ______.

The number of natural numbers, between \( 212 \) and \( 999 \), such that the sum of their digits is \( 15 \), is __________.

A person has three different bags and four different books. The number of ways, in which he can put these books in the bags so that no bag is empty, 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_______.

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 \( P \). The maximum number of points of intersection of pairs of lines form the set \( \{L_1,L_2,\ldots,L_{20}\} \) is equal to \( \_\_\_\_\_\_\_\_\_\_ \).

In an examination, there are \( 5 \) multiple choice questions with \( 3 \) choices, out of which exactly one is correct. There are \( 3 \) marks for each correct answer, \( -2 \) marks for each wrong answer and \( 0 \) mark if the question is not attempted. Then, the number of ways a student appearing in the examination gets \( 5 \) marks is _________.

If all the six digit numbers \( x_1x_2x_3x_4x_5x_6 \) with \( 0 < x_1 < x_2 < x_3 < x_4 < x_5 < x_6 \) are arranged in the increasing order, then the sum of the digits in the \( 72^{\text{th}} \) number is ________.

Let \( A=\{(a,b,c):a,b,c \\text{ are non-negative integers and } a+b+2c=22\} \). Then \( n(A) \) is equal to:

The number of seven-digit numbers, that can be formed by using the digits \( 1, 2, 3, 5 \) and \( 7 \) such that each digit is used at least once, is:

The number of \( 3 \) digit numbers, that are divisible by either \( 3 \) or \( 4 \) but not divisible by \( 48 \), is:

Let the digits \( a,b,c \) be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?

The number of elements in the set \( \{\, n\in \mathbb{N}: 10\le n\le 100 \text{ and } 3^n-3 \text{ is a multiple of } 7 \,\} \) is ________.

Let \( S \) be the set of all passwords which are six to eight characters long, where each character is either an alphabet from \( \{A,B,C,D,E\} \) or a number from \( \{1,2,3,4,5\} \) with the repetition of characters allowed. If the number of passwords in \( S \) whose at least one character is a number from \( \{1,2,3,4,5\} \) is \( \alpha \times 5^{6} \), then \( \alpha \) is equal to ________.

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:

Two players \( A \) and \( B \) play a series of games of badminton. The player, who wins \( 5 \) games first, wins the series. Assuming that no game ends in a draw, the number of ways, in which player \( A \) wins the series is ______.

The number of elements in the set \( S = \{(r,k): k \in \mathbb{Z} \text{ and } {}^{36}C_{r+1} = \dfrac{6\left({}^{35}C_{r}\right)}{k^{2}-3}\} \) is:

If \( {}^{n}P_{r} = {}^{n}P_{r+1} \) and \( {}^{n}C_{r} = {}^{n}C_{r-1} \), then the value of \( r \) is equal to:

All the letters of the word “GTWENTY” 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 “GTWENTY” is __________.

The total number of four digits numbers such that each of first three digits is divisible by the last digit, is equal to ______.

The number of \( 4 \)-digit numbers which are neither multiple of \( 7 \) nor multiple of \( 3 \) 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:

The total number of six digit numbers, formed using the digits \( 4, 5, 9 \) only and divisible by \( 6 \), is ______.

If the digits are not allowed to repeat in any number formed by using the digits \( 0, 2, 4, 6, 8 \), then the number of all numbers greater than \( 10{,}000 \) is equal to _______.

There are \( 5 \) points \( P_{1},P_{2},P_{3},P_{4},P_{5} \) on the side \( AB \), excluding \( A \) and \( B \), of a triangle \( ABC \). Similarly there are \( 6 \) points \( P_{6},P_{7},\ldots,P_{11} \) on the side \( BC \) and \( 7 \) points \( P_{12},P_{13},\ldots,P_{18} \) on the side \( CA \) of the triangle. The number of Triangles, that can be formed using the points \( P_{1},P_{2},\ldots,P_{18} \) as vertices, is:

Number of \( 4 \)-digit numbers (the repeation of digits is allowed) which are made using the digits \( 1,2,3 \) and \( 5 \), and are divisible by \( 15 \), is equal to ________.

The number of ways of giving \( 20 \) distinct oranges to \( 3 \) children such that each child gets at least one orange is ________________.

The letters of the word ‘\( \text{MANKIND} \)’ are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word ‘\( \text{MANKIND} \)’ is ______.

The number of bijective functions \( f:\{1,3,5,7,\ldots,99\}\to\{2,4,6,8,\ldots,100\} \), such that \( f(3)\ge f(9)\ge f(21)\ge \ldots \ge f(99) \), is ______________.

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

A boy needs to select five courses from \( 12 \) available courses, out of which \( 5 \) courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is _______.

The number of \( 3 \)-digit numbers, that are divisible by either \( 2 \) or \( 3 \) but not divisible by \( 7 \), is _______.

The largest natural number \( n \) such that \( 3^{n} \) divides \( 66! \) is ________.

Team \( 'A' \) consists of \( 7 \) boys and \( n \) girls and Team \( 'B' \) has \( 4 \) boys and \( 6 \) girls. If a total of \( 52 \) single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then \( n \) is equal to:

The number of sequences of ten terms, whose terms are either \( 0 \) or \( 1 \) or \( 2 \), that contain exactly five \( 1 \)s and exactly three \( 2 \)s, is equal to:

There are \( 5 \) students in class \( 10 \), \( 6 \) students in class \( 11 \) and \( 8 \) students in class \( 12 \). If the number of ways, in which \( 10 \) students can be selected from them so as to include at least \( 2 \) students from each class and at most \( 5 \) students from the total \( 11 \) students of class \( 10 \) and \( 11 \) is \( 100k \), then \( k \) is equal to ____________ .

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

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 digit should not be more than \( 8 \), is:

The total number of \( 5 \)-digit numbers, formed by using the digits \( 1, 2, 3, 5, 6, 7 \) without repetition, which are multiple of \( 6 \), is:

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 seven-digit positive integers formed using the digits \( 1 \), \( 2 \), \( 3 \) and \( 4 \) only and sum of the digits equal to \( 12 \) is ______.

The students \( s_1, s_2, \ldots, s_{10} \) are to be divided into \( 3 \) groups \( A \), \( B \) and \( C \) such that each group has at least one student and the group \( C \) has at most \( 3 \) students. Then the total number of possibilities of forming such groups is _______.

If the series \( AB \), \( BC \) and \( CA \) of a triangle \( ABC \) have \( 3 \), \( 5 \) and \( 6 \) interior points respectively, then the total number of triangles that can be constructed using these as vertices, is equal to:

The number of numbers greater than \( 5000 \), less than \( 9000 \) and divisible by \( 3 \), that can be formed using the digits \( 0, 1, 2, 5, 9 \), if the repetition of the digits is allowed, is _______.

The number of times the digit \( 3 \) will be written when listing the integers from \( 1 \) to \( 1000 \) is ____________.

The number of triplets \( (x, y, z) \), where \( x, y, z \) are distinct non negative integers satisfying \( x + y + z = 15 \), is:

Number of integral solutions to the equation \( x + y + z = 21 \), where \( x \ge 1 \), \( y \ge 3 \), \( z \ge 4 \), is equal to \( \_\_\_\_ \).

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:

Let \( A = \{2, 3, 4\} \) and \( B = \{8, 9, 12\} \). Then the number of elements in the relation \( R = \{((a_1, b_1), (a_2, b_2)) \in (A \times B, A \times B) : a_1 \text{ divides } b_2 \text{ and } a_2 \text{ divides } b_1\} \) is:

Numbers are to be formed between \( 1000 \) and \( 3000 \), which are divisible by \( 4 \), using the digits \( 1,2,3,4,5, \text{and}\ 6 \) without repetition of digits. Then the total number of such numbers is ______.

The total number of words (with or without meaning) that can be formed out of the letters of the word ‘DISTRIBUTION’ taken four at a time, is equal to _______

From a group of \( 7 \) batsmen and \( 6 \) bowlers, \( 10 \) players are to be chosen for a team, which should include atleast \( 4 \) batsmen and atleast \( 4 \) bowlers. One batsmen and one bowler who are captain and vice-captain respectively of the team should be included. Then the total number of ways such a selection can be made, is:

The number of three-digit even numbers, formed by the digits \( 0 \), \( 1 \), \( 3 \),\( 4 \), \( 6 \), \( 7 \) if the repetition of digits is not allowed, is _________.

The number of singular matrices of order \( 2 \), whose elements are from the set \( \{2,3,6,9\} \), is ____________.

The value of $\displaystyle \sum_{r=0}^{22} {}^{22}C_r \ {}^{23}C_r$ is:

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:

If \( {^{2n}C_3}:{^{n}C_3}=10:1 \), then the ratio \( (n^2+3n):(n^2-3n+4) \) 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:

The total number of three-digit numbers, with one digit repeated exactly two times, is __________.

The number of symmetric relations defined on the set \( \{ 1, 2, 3, 4 \} \) which are not reflexive is _______.

The largest value of \( n \), for which \( 40^n \) divides \( 60! \), is:

Let \( S \) denote the set of \( 4 \)-digit numbers \( abcd \) such that \( a>b>c>d \) and \( P \) denote the set of \( 5 \)-digit numbers having product of its digits equal to \( 20 \). Then \( n(S)+n(P) \) is equal to \( \_\_\_\_\_\_\_\_ \).

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JEE Main 2023 6th April Morning Shift

The number of ways of giving \( 20 \) distinct oranges to \( 3 \) children such that each child gets at least one orange is ________________.