Permutations and Combinations

The number of integers greater than 6,000 that can be formed, using the digits 3, 5, 6, 7 and 8, without repetition, is :

The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices (0, 0), (0, 41) and (41, 0), is :

If 12 identical balls are to be placed in 3 identical boxes, then the probability that one of the boxes contains exactly 3 balls is :

The number of ways of selecting 15 teams from 15 men and 15 women, such that each team consists of a man and a woman, is :

Let $A=\{x_1,x_2,\dots,x_7\}$ and $B=\{y_1,y_2,y_3\}$ be two sets containing seven and three distinct elements respectively. Then the total number of functions $f:A\to B$ that are onto, if there exist exactly three elements $x$ in A such that $f(x)=y_2$, is equal to :

If in a regular polygon the number of diagonals is 54, then the number of sides of this polygon is :

If the lengths of the sides of a triangle are decided by the three throws of a single fair die, then the probability that the triangle is of maximum area given that it is an isosceles triangle, is :

Let $n\ge2$ be an integer. Take $n$ distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining every pair of adjacent points by blue and the rest by red. If the number of red and blue line segments are equal, then the value of $n$ is

Let $n_1<n_2<n_3<n_4<n_5$ be positive integers such that $n_1+n_2+n_3+n_4+n_5=20$. Then the number of such distinct arrangements $(n_1,n_2,n_3,n_4,n_5)$ is

Coefficient of $x^{11}$ in the expansion of $(1+x^2)^4(1+x^3)^7(1+x^4)^{12}$ is

Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same number and moreover the card numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done is

Three boys and two girls stand in a queue. The probability, that the number of boys ahead of every girl is at least one more than the number of girls ahead of her, is

The sum of the digits in the unit's place of all the 4-digit numbers formed by using the numbers 3, 4, 5 and 6, without repetition, 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 set S contains 7 elements. A non-empty subset A of S and an element $x$ of S are chosen at random. Then the probability that $x\in A$ is :

8 - digit numbers are formed using the digits 1, 1, 2, 2, 2, 3, 4, 4. The number of such numbers in which the odd digits do not occupy odd places, is :

Two women and some men participated in a chess tournament in which every participant played two games with each of the other participants. If the number of games that the men played between them-selves exceeds the number of games that the men played with the women by 66, then the number of men who participated in the tournament lies in the interval :

The coefficient of $x^{1012}$ in the expansion of $(1+x^n+x^{253})^{10}$, (where $n\le22$ is any positive integer), is :

Consider the set of eight vectors $V=\left\{a\hat i+b\hat j+c\hat k:a,b,c\in\{-1,1\}\right\}$. Three non-coplanar vectors can be chosen from $V$ in $2^p$ ways. Then $p$ is

The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is