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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:

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 \( 3 \)-digit numbers, that are divisible by \( 2 \) and \( 3 \), but not divisible by \( 4 \) and \( 9 \), 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:

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:

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:

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 ______.

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:

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 words, which can be formed using all the letters of the word “DAUGHTER”, so that all the vowels never come together, 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 \( \underline{\phantom{xxxx}} \).

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

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:

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:

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:

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 singular matrices of order \( 2 \), whose elements are from the set \( \{2,3,6,9\} \), 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:

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:

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JEE Main 2025 2nd April Morning Shift

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