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

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 \( b_1 b_2 b_3 b_4 \) 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 ________.

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:

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 ________.

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 ______________.

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 _________.

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 ________.

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

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 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:

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 three-digit numbers, with one digit repeated exactly two times, is __________.

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JEE Main 2022 28th July Evening shift

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 ______________.