Permutations & Combinations Questions (855)

How many words can be formed using all the letters of the word BANANA?

How many words can be formed using all the letters of the word INDEPENDENCE?

Find the number of ways in which 5 A's and 6 B's can be arranged in a row which reads the same backwards and forwards.

The value of \({}^{50}C_4 + \displaystyle\sum_{r=1}^{6} {}^{56-r}C_3\) is:

How many numbers can be formed from the digits 1, 2, 3, 4 when repetition is not allowed?

A team of four students is to be selected from a total of 12 students. The total number of ways in which the team can be selected such that two particular students refuse to be together and other two particular students wish to be together only is equal to

If the difference of the number of arrangements of three things from a certain number of dissimilar things and the number of selections of the same number of things from them exceeds 100, then the least number of dissimilar things is

All words from MONDAY in dictionary order. Serial number of MONDAY is

In how many different ways can the first 12 natural numbers be divided into three different groups such that numbers in each group are in A.P.?

There are n straight lines in a plane, in which no two are parallel and no three pass through the same point. Their points of intersection are joined. Show that the number of new lines thus introduced is\[\frac{1}{8}n(n-1)(n-2)(n-3)\]

A guardian with $6$ wards wishes everyone of them to study either Law or Medicine or Engineering. Number of ways in which he can make up his mind with regard to the education of his wards if every one of them be fit for any of the branches to study, and atleast one child is to be sent in each discipline is:

Consider the letters of the word MATHEMATICS. Possible number of words taking all letters at a time such that in each word both M's are together and both T's are together but both A's are not together is

How many new words can be formed using all the letters of the word "MEDITERRANEAN", if vowels and consonants occupy the same relative positions?

m equispaced horizontal lines are intersected by n equispaced vertical lines. If the distance between two successive horizontal lines is same as that between two successive vertical lines, then find the number of squares formed by the lines if \(m < n\).

If the total number of non-decreasing functions defined from $f : \{1,2,3,4,5\} \to \{1,2,3,4,5,6,7,8,9\}$ is $m$ then $\frac{m}{143}$ is equal to ______.

Straight lines are drawn by joining \(m\) points on a straight line to \(n\) points on another line. Then excluding the given points, the number of point of intersections of the lines drawn is (no two lines drawn are parallel and no three lines are concurrent)

A rack has 5 different pairs of shoes. The number of ways in which 4 shoes can be chosen from it, so that there will be no complete pair is:

A regular polygon of 10 sides is constructed. In how many ways can 3 vertices be selected so that no two vertices are consecutive?

All possible 6 digit numbers, in each of which the digits occur in non increasing order (From left to right e.g. 877550) are written as a sequence in increasing order. Find the 2005th number in this sequence.

From 4 men and 6 ladies a committee of 5 is to be selected. The number of ways in which the committee can be formed so that men are in majority, is:

If all the words, with or without meaning, are written using the letters of the word QUEEN and are arranged as in English dictionary, then the position of the word QUEEN is

Find the three-digit odd numbers that can be formed by using the digits 1, 2, 3, 4, 5, 6 when the repetition is allowed.

In how many ways can \(2t+1\) identical balls be placed in three distinct boxes so that any two boxes together will contain more balls than the third?

The total number of three-letter words that can be formed from the letter of the word SAHARANPUR is equal to

Find the number of ways of selecting 10 objects from 42 objects of 21 objects are identical and remaining objects are distinct.

Four buses run between Bhopal and Gwalior. If a man goes from Gwalior to Bhopal by a bus and comes back to Gwalior by another bus, find the total possible ways.

Numbers greater than 1000 but not greater than 4000 which can be formed with the digits 0, 1, 2, 3, 4 (repetition of digits is allowed) are

How many natural numbers are there lying between 20,000 and 60,000, the sum of digits being even?

Roorkee University has to send 10 professors to 5 centers for its entrance examination, 2 to each center. Two of the centers are in Roorkee and the others are outside. Two of the professors prefer to work in Roorkee while three prefer to work outside. In how many ways can this be made if the preferences are to be satisfied?

Consider the equation \(\dfrac{2}{x} + \dfrac{5}{y} = \dfrac{1}{3}\), where \(x, y \in \mathbb{N}\). Find the number of solutions of the equation.

The total number of different ways a grandfather along with two of his grandsons and four grand daughters can be seated in a line for a photograph so that he is always in the middle and the two grandsons are never adjacent to each other:

Consider a $6 \times 6$ square which is dissected into 9 rectangles by lines parallel to its sides such that all the rectangles have integral sides. What is the minimum number of congruent rectangles?

There are 12 balls numbered from 1 to 12. The number of ways in which they can be used to fill 8 places in a row so that the balls are with numbers in ascending or descending order is equal to

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

Eighteen guests have to be seated, half on each side of a long table. Four particular guests desire to sit on one particular side and three others on the other side. Determine the number of ways in which the sitting arrangements can be made.

Consider a polygon of $k$ sides. If the number of triangles that can be drawn taking vertices of these polygons as vertices of triangles and no sides of triangles is common with any sides of the polygon is 50 then $k$ is_____.

A man has 7 relatives, 4 of them are ladies and 3 gentlemen; his wife has 7 relatives, 3 of them are ladies and 4 gentlemen. In how many different ways can they invite a dinner party of 3 ladies and 3 gentlemen so that there are 3 of the man's relatives and 3 of the wife's relatives?

A dictionary is printed consisting of 7-lettered words that can be made with the letters of the word CRICKET. If the words are printed in the alphabetic order, as in an ordinary dictionary, find the position of the word CRICKET in that dictionary.

Six persons $A, B, C, D, E$ and $F$ are to be seated at a circular table. The number of ways this can be done if $A$ must have either $B$ or $C$ on his right and $B$ must have either $C$ or $D$ on his right is:

There are m men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by 84, then the value of m is __________.

Find the number of ways in which 6 men and 5 women can dine at a round table if no two women are to sit together.

The set \(S = \{1, 2, 3, \ldots, 12\}\) is to be partitioned into three sets \(A\), \(B\), \(C\) of equal size. Thus, \(A \cup B \cup C = S\), \(A \cap B = B \cap C = A \cap C = \phi\). The number of ways to partition \(S\) is

At IITD, roll number of $N$ students are given from 1 to $N$. Three students are selected from these $N$ students such that their roll numbers are not consecutive, the total number of ways this selection can be done is 10, then N is equal to_____.

In a class of 10 students if two prizes (1st and 2nd) has to be given in three subjects. Physics, Chemistry & Mathematics and this can be done in $k$ ways, then $\frac{k}{1000}$ is_____.

The number of non-negative integral solutions of $x+y+z \leq n$ where $n \in \mathbb{N}$ is:

Find the number of non-negative integral solutions of \(x_1 + x_2 + x_3 + 4x_4 = 20\).

If $\lambda$ be the number of 3-digit numbers are of the form $xyz$ with $x < y, z < y$ and $x \neq 0$, the value of $\lambda$ is_____.

Let \(A = \{1, 2, 3, 4, 5, 6, 7\}\). The number of surjective functions defined from \(A\) to \(A\) such that \(f(i) = i\) for at least four values of \(i\) from \(i = 1, 2, \ldots, 7\) is:

Find the number of ways of selecting 10 balls out of an unlimited number of identical white, red, and blue balls.

In how many ways can a committee of 10 be selected with at least 4 women and 4 men from 9 women and 8 men if(i) Ms X refuses to work with Mr Y(ii) Ms X and Mr Y insist to work together?