5 men and 4 women are seated in a row. If the number of arrangements in which one particular man and one particular woman are together is $\alpha$ and the number of arrangements in which those two are not together is $\beta$, then $\alpha$ : $\beta=$

If a team of 4 persons is to be selected out of 4 married couples to play mixed doubles- tennis game, then the number of ways of forming a team in which no married couple appears is

An eight digit number divisible by 9 is to be formed using digits from 0 to 9 without repeating the digits. The number of ways in which this can be done is

A string of letters is to be formed by using 4 letters from all the letters of the word "MATHEMATICS". The number of ways this can be done such that two letters are of same kind and the other two are of different kind is

The number of integers greater than 6000 that can be formed by using the digits $0,5,6,7,8$ and 9 without repetition is

The number of ways of dividing 15 persons into 3 groups containing 3,5 and 7 persons so that two particular persons are not included into the 5 persons groups is

The number of integers between 10 and 10,000 such that in every integer every digit is greater than its immediate preceeding digit, is

The number of ways in which a cricket team of 11 members can be formed out of 6 batsmen, 6 bowlers, 4 all-rounders and 4 wicket keepers by selecting atleast 4 batsmen, atleast 3 bowlers, atleast 2 all-rounders and only one wicket keeper is

If all possible 4 -digit numbers are formed by choosing 4 different digits from the given digits $1,2,3,5,8$ then the sum of all such 4 -digit numbers is

The number of ways in which a committee of 7 members can be formed from 6 teachers, 5 fathers and 4 students in such a way that at least one from each group is included and teachers form the majority among them, is

If 3 sisters and 8 brothers are together playing a game, then the number of ways in which all the sisters and brothers are to be seated around a circle such that all the three sisters are not seated together is

Out of 8 students in a classroom, 4 of them are chosen and they are arranged around a table.

If the remaining 4 are arranged in a row, then the total number of arrangements that can be made with those 8 students is

Three letters are chosen at random from the letters of the word VARIABLE and all possible three letter words (with or without meaning) are formed with them.

Then, the probability of getting a three letter word having a consonent as its middle letter is

If ${ }^{27} P_{r+7}=7722{ }^{25} P_{(r+4)}$, then $r=$

If the number of diagonals of a regular polygon is 35 , then the number of sides of the polygon is

If four letters are chosen from the letters of the word ASSIGNMENT and are arranged in all possible ways to form 4 letter words (with or without meaning), then total number of such words that can be formed is

All the letters of the word LETTER are arranged in all possible ways and the words (with or without meaning) thus formed are arranged in dictionary order.

Then, the rank of the word TETLER is

5-digit numbers are formed by using the digits $0,1,2$, $3,5,7$ without repetetion and all of them are arranged in ascending order. Then, the rank of the number 70513 is

The number of divisors of 7 ! is

If all the letters of the word COMBINATION are arranged in all possible ways to form 11 letter words (with or without meaning), then the number of words among them in which $C$ and $N$ occupy the end positions and no vowel appears exactly in the middle position is

The number of ways of distributing 3 dozen fruits (no two fruits are identical) to 9 persons such that each gets the same number of fruits is

If $\binom{p}{q}={ }^p C_q$ and $\sum\limits_{i=0}^m\binom{10}{i}\binom{20}{m-i}$ is maximum, then $m=$

The number of all possible positive integrals solutions of the equation $x y z=30$ is

The number of all five letter words (with or without meaning) having atleast one repeated letter than can be formed by using the letters of the word INCONVENIENCE is

The number of ways of arranging all the letters of the word PERFECTION such that there must be exactly two consonants between any two vowels is

There were two women participating with some men in a chess tournament. Each participant played two games with the other. The number of games that the men played between themselves is 66 more than that of the men played with the women. Then, the total number of participants in the tournament is

If there are 6 alike fruits, 7 alike vegetables and 8 alike biscuits, then the number of ways of selecting any number of things out of them such that at least one from each category is selected, is

All the letters of the word 'TABLE' are permuted and the strings of letters (may or may not have meaning) thus formed are arranged in dictionary order. Then, the rank of the word 'TABLE' counted from the rank of the word 'BLATE' is

$\text { If } 10{ }^n C_2=3^{n+1} C_3 \text {, then the value of } n \text { is }$

There are 10 points in a plane, out of these 6 are collinear. If $N$ is the total number of triangles formed by joining these points, then $N=$

In an examination, the maximum marks for each of three subjects is $n$ and that for the fourth subject is $2 n$. The number of ways in which candidates can get $3 n$ marks is

If a set $A$ has $m$-elements and the set $B$ has $n$-elements, then the number of injections from $A$ to $B$ is

In how many ways can the letters of the word "MULTIPLE" be arranged keeping the position of the vowels fixed?

A natural number $n$ such that $n!$ ends in exactly 1000 zeroes is

The total number of permutations of $n$ different things taken not more than $r$ at a time, when each thing may be repeated any number of times is

How many chords can be drawn through 21 points on a circle?

If a polygon of $n$ sides has 560 diagonals, then $n=$

A person writes letters to 6 friends and addresses the corresponding envelopes. In how many ways can the letters be placed in the envelopes so that at least two of them are in the wrong envelopes? Notation $D_n=n!\left(\sum_\limits{i=0}^n \frac{(-1)^i}{i!}\right)$

A set contains 11 elements. The number of subsets of the set which contain at most 5 elements is

The value of ${ }^6 P_4+4 \cdot{ }^6 P_3$ is

The number of ways in which 3 boys and 2 girls can sit on a bench so that no two boys are adjacent is

In how many ways can 5 balls be placed in 4 tins if any number of balls can be placed in any tin?

For $1 \leq r \leq n, \frac{1}{r+1}\left\{{ }^n P_{r+1}-{ }^{(n-1)} P_{r+1}\right\}$ is equal to

In how many ways 4 balls can be picked from 6 black and 4 green coloured balls such that at least one black ball is selected?

In how many ways can 9 examination papers be arranged so, that the best and the worst papers are never together?

If a person has 3 coins of different denominations, the number of different sums can be formed is

There are 7 identical white balls and 3 identical black balls. The number of distinguishable arrangements in a row of all the balls, so that no two black balls are adjacent is

The number of ways of distributing eight identical rings to three different girls so that every girl gets at least one ring is

If the letters of the word REGULATIONS be arranged in such a way that relative positions of the letters of the word GULATIONS remain the same, then the probability that there are exactly 4 letters between R and E is