The number of positive integral solution of $\frac{1}{x}+\frac{1}{y}=\frac{1}{2025}$ is

The number of positive integral solutions of $x y z=60$ is

5 boys and 5 girls have to sit around a table. The number of ways in which all of them can sit so that no two boys and no two girls are together is

All possible words (with or without meaning) the contain the word 'GENTLE' are formed using all the letters of the word 'INTELLIGENCE'. Then, the number of words in which the word 'GENTLE' appears among the first nine positions only is

$ { }^{20} P_5-{ }^{19} P_5= $

If all the letters of the word ACADEMICIAN are permuted in all possible ways, then the number of permutations in which no two $A^{\prime} s$ are together and all the consonants are together is

The number of all possible three letter words that can be formed by choosing three letters from the letters of the word FEBRUARY so that a vowel always occupies the middle place is

The number of ways in which 6 boys and 4 girls can be arranged in a row such that between any two girls there must be exactly 2 boys is

There are 15 stations on a train route and the train has to be stopped at exactly 5 stations among these 15 stations. If it stops at atleast two consecutive stations, then the number of ways in which the train can be stopped is

Number of all possible ways of distributing eight identical apples among three persons is

Number of all possible words (with or without meaning) that can be formed using all the letters of the word CABINET in which neither the word CAB nor the word NET appear is

The number of non-negative integral solutions of the equation $x+y+z+t=10$ when $x \geq 2, z \geq 5$ is

The number of integers lying between 1000 and 10000 such that the sum of all the digits in each of those numbers becomes 30 is

If all the letters of the word MOST are permuted and the words (with or without meaning) thus obtained are arranged in the dictionary order, then the rank of the words STOM when counted from the rank of the word MOST, is

A student has to answer a multiple-choice question having 5 alternatives in which two or more than two alternatives are correct. Then, the number of ways in which the student can answer that question is

If all the letters of the word 'HANDLE' are permuted in all possible ways and the words (with or without meaning) thus formed are arranged in dictionary order, then the rank of the word 'HELAND' is

The number of odd numbers greater than 600000 that can be formed by using the digits $3,6,7,8,9,0$ without repetition is

There are three sections in a question paper, each section containing 4 questions. If a candidate has to answer only 5 questions from this paper without leaving any section, then the number of ways in which a candidate can make the choice of questions is

The number of ways in which 6 men and 4 women can be seated around a table, so that a particular man and a particular woman never sit adjacent to each other is

The number of diagonals of a polygon is 35 . If $A$ and $B$ are two distinct vertices of this polygon, then the number of all those triangles formed by joining three vertices of the polygon having $A B$ as one of its sides is

There are 10 points in a plane, of which no three points are collinear except 4. Then, the number of distinct triangles that can be formed by joining any three points of these ten points, such that at least one of the vertices of every triangle formed is from the given 4 collinear points is

A student is asked to answer 10 out of 13 questions in an examination such that he must answer atleast four questions from the first five questions. Then, the total number of possible choices available to him is

All the letters of the word 'INDEED' are taken and permuted in all possible ways to form distinct 6 letter strings (words with or without meaning). If they are listed in dictionary order, then the rank position of the string 'NIDDEE' is

All possible 5-digit numbers each having 5 distinct digits are formed using the digits $1,2,3,5,6,8$. Among them, the number of numbers which are divisible by 3 but not by 6 is

The total number of ways of forming a committee of 5 members out of 7 Indians, 6 Americans, 5 Russians and 4 Australians, so that every committee contains atleast one member from each country is

If $n, r$ are two positive integers such that $1 \leq r

The number of ways in which $n$ boys and $n$ girls can be arranged in a row such that all the boys are together and all the girls are also together is equal to

Among the positive divisors of the number 12600 , if $n_1$ is the number of divisors which are multiples of 3 and $n_2$ is the number of divisors which are multiples of 14 , then $n_1+n_2=$

All the letters of the word 'MOTHER' are written in all possible ways and the strings of letters (with or without meaning), so formed are written as in a dictionary order. Then, the position of the word 'THROEM' is

A student is allowed to select at most $n$ books from a collection of ( $2 n+1$ ) books. If the total number of ways in which he can select at least one book is 255 , then the value of $n$ is

If ${ }^m P_r-{ }^{(m-1)} p_r=a \cdot{ }^{(m-1)} P_s$, then $a-s=$

The total number of ways of selecting 4 letters from all the letters of the word TSEAMCET is

Let $a, b, c \in N$ and $a+b+c=5$. Let $L, M$ be the least and greatest values of $2^a 3^b 5^c$, respectively. Then $M-L=$

The number of positive divisors of 360 which are multiples of 3 is

The number of ways of arranging the letters of the word LINEAR so that the letters N and R do not come together and E and A come together is

15 lines are concurrent at a point $P$. A line $L$ is not passing through $P$ intersects all the 15 lines and forms triangles with them. Then, the number of triangles having $L$ as one of its side is

Let $N$ be the set of positive integers. The number of distinct triplets $(x, y, z)$ satisfying $x, y, z \in N, x

A question paper has 3 parts and each part contains 4 questions. The number of different ways in which a candidate can answer 8 questions choosing at least two from each part is

$a, b, c$ are three particular speakers among the 10 speakers of a meeting. The number of ways of arranging all 10 speakers on the dias in a row so that all the three speakers $a, b, c$ do not sit together is

The exponent of 6 in 72 ! is

The number of 3-digit odd numbers divisible by 3 that can be formed using the digits $1,2,3,4,5,6$ when repetition is not allowed, is

$ \text { Match the items of List-I to the items of List-II } $

For $n=1,2,3, \ldots .50$, let

$ A=\left\{a_n / a_n=\left\{\begin{array}{ll} (-1)^{\frac{n}{2}}\left(\frac{n}{2}\right), & \text { if } n \text { is even } \\ (-1)^{\frac{n-1}{2}}\left(\frac{n-1}{2}\right), & \text { if } n \text { is odd } \end{array}\right\}\right\} $

and $B$ is the set of all distinct elements of $A$. The number of permutations all the elements of set $B$ such that even integers are in increasing order, is

If $\alpha$ represents the number of arrangements of $p$ men and $q$ women in a row such that all men are together and $\beta$ represents the number of circular arrangements of the same people with the same condition, then $\alpha: \beta$ is

Consider the following statements:

I. The number of positive integral solutions of $x_1+x_2+x_3+x_4=10$ is 286 .

II. If $25!=10^n \times k,(k \in \mathbf{N})$, then $n=6$

Which one of the following options is true?

A student is allowed to select at least $(n+1)$ books but not all books from a collection of ( $2 n+1$ ) books. If the total number of ways in which he can select these books is 255 , then the number of books in that collection is

If $x$ and $y$ represent the number of arrangements of the letters of word ATRAPATRAM such that (i) all A's are together and (ii) no two A's are together respectively, then $x+y$

Numbers between 1 and 10,000 are formed using the digits 2 and 3 only once and the digit 4 twice. If the numbers thus formed are arranged in increasing order and $x, y$ represent the ranks of 4324 and 324 respectively then $x-y=$

The total number of three digit and five digit integers which can be formed by using the digits $0,1,2,3,4,5$ but using each digit not more than once in each number is

At an election a voter may vote for any number of candidates not exceeding the number to be elected. If 4 candidates are to be elected out of the 12 contested in the election and voter votes for at least one candidate, then the number of ways in which a voter can vote is