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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:

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 ______.

\( 60 \) 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 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:

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 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:

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

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:

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:

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 ______.

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 \( \_\_\_\_\_\_\_\_\_\_ \).

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 __________.

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:

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

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:

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 _______

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

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JEE Main 2024 27th Jan Evening Shift

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