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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:

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,

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 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:

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 of \( 4 \)-letter words, with or without meaning, which can be formed using the letters PQRPQRSTUVP, 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:

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 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 ways, in which \( 16 \) oranges can be distributed to four children such that each child gets at least one orange, 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 _________.

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 \( A=\{(a,b,c):a,b,c \\text{ are non-negative integers and } a+b+2c=22\} \). 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:

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:

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 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 2026 24th Jan Evening Shift

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