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The number of \( 4 \)-letter words, with or without meaning, each consisting of \( 2 \) vowels and \( 2 \) consonants, which can be formed from the letters of the word UNIVERSE without repetition is ________ .

The sum of all the four-digit numbers that can be formed using all the digits \( 2 \), \( 1 \), \( 2 \), \( 3 \) is equal to ______.

The number of permutations, of the digits \( 1,2,3,\ldots,7 \) without repetition, which neither contain the string \( 153 \) nor the string \( 2467 \), is ____________.

The number of five digit numbers, greater than \( 40000 \) and divisible by \( 5 \), which can be formed using the digits \( 0, 1, 3, 5, 7 \) and \( 9 \) without repetition, is equal to:

The number of seven digits odd numbers, that can be formed using all the seven digits \( 1, 2, 2,2, 3, 3, 5 \) is ________.

Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is \( 840 \), then the total numbers of persons, who participated in the tournament, is ________.

The total number of three-digit numbers, divisible by \( 3 \), which can be formed using the digits \( 1,3,5,8 \), if repetition of digits is allowed, is:

Let \( S=\{\,1,2,3,5,7,10,11\,\} \). The number of non-empty subsets of \( S \) that have the sum of all elements a multiple of \( 3 \), is ______.

All words, with or without meaning, are made using all the letters of the word MONDAY. These words are written as in a dictionary with serial numbers. The serial number of the word MONDAY is :

The number of numbers, strictly between \( 5000 \) and \( 10000 \) can be formed using the digit \( 1,3,5,7,9 \) without repetition, is:

In an examination, \( 5 \) students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is ______.

The number of integers, greater than \( 7000 \) that can be formed, using the digits \( 3, 5, 6, 7, 8 \) without repetition, is:

The total number of \( 4 \)-digit numbers whose greatest common divisor with \( 54 \) is \( 2 \), is ______.

Eight persons are to be transported from city \( A \) to city \( B \) in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is:

The number of ways of selecting two numbers \( a \) and \( b \), \( a\in\{2,4,6,\ldots,100\} \) and \( b\in\{1,3,5,\ldots,99\} \) such that \( 2 \) is the remainder when \( a+b \) is divided by \( 23 \) is:

If the number of words, with or without meaning, which can be made using all the letters of the word MATHEMATICS in which C and S do not come together, is \( (6!){k} \) then \( k \) is equal to:

The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is:

Suppose Anil’s mother wants to give \( 5 \) whole fruits to Anil from a basket of \( 7 \) red apples, \( 5 \) white apples and \( 8 \) oranges. If in the selected \( 5 \) fruits, at least \( 2 \) oranges, at least one red apple and at least one white apple must be given, then the number of ways, Anil’s mother can offer \( 5 \) fruits to Anil is ____.

A person forgets his \( 4 \)-digit ATM pin code. But he remembers that in the code all the digits are different, the greatest digit is \( 7 \) and the sum of the first two digits is equal to the sum of the last two digits. Then the maximum number of trials necessary to obtain the correct code is \( \_\_\_\_\_\_\_\_ \).

Total number of \( 3 \)-digit numbers that are divisible by \( 6 \) and can be formed by using the digits \( 1 \), \( 2 \), \( 3 \), \( 4 \), \( 5 \) with repetition, is _________.

The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is:

If $^{2n+1}P_{n-1} : {^{2n-1}P_n} = 11 : 21$, then $n^2 + n + 15$ is equal to ________.

If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial numbers, then the serial number of the word THAMS is:

Five digit numbers are formed using the digits \( 1, 2, 3, 5, 7 \) with repetitions and are written in descending order with serial numbers. For example, the number \( 77777 \) has serial number \( 1 \). Then the serial number of \( 35337 \) is ________.

The number of \( 9 \)-digit numbers, that can be formed using all the digits of the number \( 123412341 \) so that the even digits occupy only even places, is ______.

Let \( 5 \) digit numbers be constructed using the digits \( 0,2,3,4,7,9 \) with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number \( 42923 \) is_______.

If all the six digit numbers \( x_1x_2x_3x_4x_5x_6 \) with \( 0 < x_1 < x_2 < x_3 < x_4 < x_5 < x_6 \) are arranged in the increasing order, then the sum of the digits in the \( 72^{\text{th}} \) number is ________.

The number of \( 3 \) digit numbers, that are divisible by either \( 3 \) or \( 4 \) but not divisible by \( 48 \), is:

Let the digits \( a,b,c \) be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?

The number of elements in the set \( \{\, n\in \mathbb{N}: 10\le n\le 100 \text{ and } 3^n-3 \text{ is a multiple of } 7 \,\} \) is ________.

The number of ways, in which \( 5 \) girls and \( 7 \) boys can be seated at a round table so that no two girls sit together, is:

The total number of six digit numbers, formed using the digits \( 4, 5, 9 \) only and divisible by \( 6 \), is ______.

Number of \( 4 \)-digit numbers (the repeation of digits is allowed) which are made using the digits \( 1,2,3 \) and \( 5 \), and are divisible by \( 15 \), is equal to ________.

The number of ways of giving \( 20 \) distinct oranges to \( 3 \) children such that each child gets at least one orange is ________________.

The number of words, with the without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is _______.

A boy needs to select five courses from \( 12 \) available courses, out of which \( 5 \) courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is _______.

The number of \( 3 \)-digit numbers, that are divisible by either \( 2 \) or \( 3 \) but not divisible by \( 7 \), is _______.

The largest natural number \( n \) such that \( 3^{n} \) divides \( 66! \) is ________.

The number of seven-digit positive integers formed using the digits \( 1 \), \( 2 \), \( 3 \) and \( 4 \) only and sum of the digits equal to \( 12 \) is ______.

The number of triplets \( (x, y, z) \), where \( x, y, z \) are distinct non negative integers satisfying \( x + y + z = 15 \), is:

Number of integral solutions to the equation \( x + y + z = 21 \), where \( x \ge 1 \), \( y \ge 3 \), \( z \ge 4 \), is equal to \( \_\_\_\_ \).

Let \( A = \{2, 3, 4\} \) and \( B = \{8, 9, 12\} \). Then the number of elements in the relation \( R = \{((a_1, b_1), (a_2, b_2)) \in (A \times B, A \times B) : a_1 \text{ divides } b_2 \text{ and } a_2 \text{ divides } b_1\} \) is:

The value of $\displaystyle \sum_{r=0}^{22} {}^{22}C_r \ {}^{23}C_r$ is:

If \( {^{2n}C_3}:{^{n}C_3}=10:1 \), then the ratio \( (n^2+3n):(n^2-3n+4) \) is:

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JEE Main 2023 1st Feb Evening Shift

Number of integral solutions to the equation \( x + y + z = 21 \), where \( x \ge 1 \), \( y \ge 3 \), \( z \ge 4 \), is equal to \( \_\_\_\_ \).