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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 23rd Jan Evening Shift

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