Permutations and Combinations

A polygon has 14 diagonals then the number of sides are

The number of four-letter words can be formed using the letters of the word DICTIONARY is

The number of words that can be formed using the letters of the "PETROL" such that the words do not have "P" in the first position, is

The no. of different ways the letters of the word "DETAIL" can be arranged in such a way that the vowels can occupy only the odd position is

The value of $N$ in $N! + \frac{1}{7!} = \frac{1}{8!}$ is

There are ten flights operating between city A and city B. The number of ways in which a person can travel from city A to city B and return by different flight is:

$^nP_3 = 2 \cdot ^nP_2$. Find $n$.

If $^nP_1 = 20 \cdot ^nP_0$, where $P$ denotes the number of permutations, then $n$ is:

Eight chairs are numbered from 1 to 8. Two women and three men are to be seated by allowing one chair for each. First, the women choose the chairs from the chairs numbered 1 to 4 and then men select the chairs from the remaining. The number of possible arrangements is:

If $^nP_2 = 12$, then the value of $n$ is

How many numbers can be formed with the help of 2, 3, 4, 5, 6, 1 which are not divisible by 5, given that it is a five-digit no. and digits are not repeating?

How many four-digit odd numbers can be formed with digits 0, 1, 2, 3, 4, 7 and 8?

The number of words from the letters of the word BHARAT, in which B and H will never come together, is

In how many different ways can the letters of the word 'DETAIL' be arranged so that the vowels occupy only the odd positions?

A person can go from place 'A' to 'B' by 11 different modes of transport but is allowed to return to 'A' by any mode other than the one earlier. The number of different ways in which the entire journey can be completed is:

If a man travels from place A to B in 10 ways then by how many ways can he come back by another train?

If $\frac{n!}{10(n-1)!} = \frac{(n-1)!}{(n-3)!}$ find 'n'.

Which of the following is a correct statement.

If $^nP_2 = 20 \cdot ^nP_0$, then the value of $n$ is given by:

8 people are seated in a row in a meeting among them the president and vice president are to be seated always in the center. What is the arrangement?

How many no. of seven-digit numbers which can be formed from the digits 3, 4, 5, 6, 7, 8, 9 no digits being repeated are not divisible by 5?

Three girls and five boys are to be seated in a row so that no two girls sit together. Total no. of ways of this arrangement are:

How many different groups of 3 people can be formed from a group of 5 people?

'$n$' locks and '$n$' corresponding keys are available, but the actual combination is not known. The maximum number of trials that are needed to assign the keys to the corresponding locks is:

Six boys and five girls are to be seated for a photograph in a row such that no two girls sit together and no two boys sit together. Find the number of ways in which this can be done.

The number of ways 4 boys and 4 girls can be seated in a row so that they are alternate is:

If $^nP_r = 12 \times ^nP_{r-1}$, then $r$ is equal to

In how many different ways can the letters of the word 'SOFTWARE' be arranged so that the vowels always come together?

In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together?

Find 'n' if $^nP_2 = 72$.

In a class of 4 boys and 3 girls, they are required to sit in a row in such a way that no two girls can sit together. Compute, in how many different ways they can sit together.

Find the value of n if $(n+1)! = 42 (n-1)!$

If $^nP_6 = 336$ and $^nC_r = 56$, then n and r will be

In a lawn different ways can four persons stand in a line for a group photograph.

If $^nP_r = 720$ and $^nC_r = 120$ then value of $r$ is

If $^nP_3 : ^nP_2 = 3 : 1$, then value of n is

If $^nP_4 = 20 ^nP_2$ then the value of 'n' is _______.

If two letters are taken at random from the word HOME, what is the Probability that none of the letters would be vowels?

In how many ways the letters of the word 'ARRANGE' be arranged?

In how many ways can the letters of the word 'STRANGE' be arranged so that the vowels never come together?

The number of ways the letters of the word 'COMPUTER' can be rearranged is

How many ways can be letters of the word 'FAILURE' be arranged so that the consonants may occupy only odd places?

In how many ways can the letters of the word 'FAILURE' be arranged so that the consonants may occupy only odd positions?

The number of words that can be formed out of the letters of the word "ARTICLE" so that vowels occupy even places is

$^nP_3 : ^nP_2 = 2 : 1$

The Sum of all the 4 digits numbers that can be formed with the digits 3,4,5,5 is

Find the number of even numbers greater than 100 that can be formed with the digits 0,1,2,3.

How many numbers can be formed with the help of 2, 3, 4, 5, 6, 1 which is not divisible by 5, given that it is a five-digit number and digits are not repeating?

If four words are taken with or without meaning from the word 'LOGARITHAM' without repetition. How many words will be formed?

The number of ways 5 boys and 5 girls can be seated at a round table, so no two boys are adjacent is:

In how many ways can the crew of an eight seated boat be arranged so that 3 of crew can row only on a stroke side and 2 on the other side?

How many 3 digit odd no. can be formed using the digits 5, 6, 7, 8, 9 if the digits can be repeated?

In how many ways can 5 boys and 3 girls sit in a row so that no two girls are together

In how many ways the letters of the word "STADIUM" be arranged in such a way that the vowels all occur together?

How many ways can 5 different trophies can be arranged on a shelf if one particular trophy must always be at the middle?

A box contains 3 pink caps, 2 purple caps and 4 orange caps. In how many ways the caps can be arranged so that the caps of the same colour are together. (Assume all caps of same colour are not identical)

How many different words can be formed with the letters of the word "LIBERTY"?

The number of ways in which 8 examination papers be arranged so that the best and worst papers never come together

5 persons are sitting in a round table in such way that Tallest Person is always on the right side of the shortest person; the number of such arrangements is

An examination paper with 10 questions consists of 6 questions in Algebra and 4 questions in Geometry. At least one question from each section is to be attempted. In how many ways can this be done?

If 12 school teams are participating in a quiz contest, then the number of ways the first, second and third positions may be won is

A question paper contains 6 questions, each having an alternative. The number of ways an examiner can answer one or more questions is

A bag contains 4 red, 3 black and 2 white balls. In how many ways 3 balls can be drawn from this bag so that they include at least one black ball?

If $^nP = 720$ and $^nC = 120$, then $n$ is

$If these are 40 guests in a party. If each guest takes a shake hand with all the remaining guests. Then the total number of hands shake is _________.$

In how many ways can 4 people be selected at random from 6 boys and 4 girls if there are to be exactly 2 girls?

A fruity basket contains 7 apples, 6 bananas, and 4 mangoes. How many selections of 3 fruits can be made so that all 3 are apples?

Out of 7 boys and 4 girls, a team of a debate club of 5 is to be chosen. The number of teams such that each team includes at least one girl is:

In how many ways can the letters of the word "ALGEBRA" be arranged without changing the relative order of the vowels?

In how many ways can the letters of the word "DIRECTOR" be arranged so that the three vowels are never together?

How many words can be formed with the letters of the word 'ORIENTAL' So that A and E always occupy odd places:

In how many ways can a party of 4 men and 4 women be seated at a circular table, so that no two women are adjacent?

The number of ways of 4 boys and 3 girls are to be seated for photograph in a row alternatively.

The number of 3-digit odd numbers can be formed using the digits 5,6,7,8,9. If repetition is allowed?

How many numbers of 3 digits can be made by using digits 3, 5, 6, 7 and 8 no. digit being repeated.

In how many ways of the word "MATHEMATICS" be arranged so that the vowels always occur together?

The number of words from the letters of the word BHARAT, in which B and H will never come together is

The value of $N$ in $\frac{1}{7!} + \frac{1}{8!} = \frac{N}{9!}$ is

There are 5 books on English, 4 books on Tamil and 3 books on Hindi. In how many ways can these books be placed on a shelf if the books on the same subjects are to be together?

Word 'REGULATION' is arranged without repetition. Find the probability that the vowels come at odd places.

The letters of the word VIOLENT are arranged so that the vowels occupy even place only. The number of permutations is:

A garden having 6 tall trees in a row. In how many ways 5 children stand, one in a gap between the trees in order to pose for a photograph?

Find the number of arrangements in which the letters of the word 'MONDAY' be arranged so that the words thus formed begin with 'M' and do not end with 'N'.

A room has 10 doors. In how many ways can a man enter the room by one door and come out by a different door.

In how many ways can 5 boys and 3 girls sit in a row so that no two girls are together?

How many ways can 5 different trophies can be arranged on a shelf if one particular trophy must always be in the middle?

If $\frac{1}{9!} + \frac{1}{10!} = \frac{x}{11!}$. The value of $x$ is

The value of $^n P_r + r.^{n} P_{r-1}$ is

From a group of 8 men and 4 women, 4 persons are to be selected to form a committee so that at least 2 women are there on the committee. In how many ways can it be done?

A business houses wishes to simultaneously elevate two of its six branch heads. In how many ways can these elevations take place?

$7 boys and 4 girls from which a team of 5 is to be selected, each team should have atleast one girl can be done in _________ ways$

If $^{1000}C_x = ^{999}C_3 + ^{999}C_y$, find $x$:

If $^{11}C_x = ^{11}C_{4x-1}$ and $x \neq 4$, then value of $^x C_x$.

$^nC_p + 2 \cdot ^nC_{p-1} + ^nC_{p-2} =?$

There are 5 questions each have four options. In how many different ways can we answer the questions?

The number of triangle that can be formed by choosing the vertices from a set of 12 points, seven of which lie on the same straight line, is:

If there are 6 points in a line and 4 points in another line. Find the number of parallelogram formed?

There are 20 points in a plane area. How many triangles can be formed by these points if 5 points are collinear?

If $^nP_r = 3024$ and $^nC_r = 126$, then find $n$ and $r$?

A committee of 3 women and 4 men is to be formed out of 5 women and 7 men. Mrs. X can refuse to serve in a committee in which Mr. Y is a member. The no. of such committees can be:

In the next world cup of cricket, there will be 12 teams divided equally into two equal groups. Team of each group will play a match against other teams of the group. From each group, 3 top teams will qualify for next round. In this round, each team will play against each other. Four top teams of this round will qualify for semifinals and play against each other and then two top teams will go to final where they play the best of three matches. How much minimum number of matches in the next world cup will be?

If $^{10}C_n = ^{10}C_{n-4}$, then $n$ is equal to:

A selection is to be made for one post of Principal and two posts of Vice-Principal. Amongst the six candidates called for the interview, only two are eligible for the post of Principal, while they all six are eligible for the post of Vice-Principal. The no. of possible combinations for the selection is:

How many total combinations can be formed of 8 different counters marked as 1, 2, 3, 4, 5, 6, 7, and 8, taking 4 counters at a time and there being at least one odd and one even numbered counter in each combination?

In a party every person shakes hands with every other person. If there are 105 handshakes in total, find the number of persons in the party.

In how many ways 3 prizes out of 5 can be distributed amongst 3 brothers equally

A Company wishes top simultaneously promotes three of its 8 department assistant mangers. In how many ways these promotions can take place?

If $^nP = 336$ and $^nC = 56$, then $n$ and $r$ will be

$^nP_r = 720$ and $^nC_r = 120$ then value of $r$ is

In how many ways can a group of 3 ladies and four gents be chosen from 8 ladies and 7 gents?

A box contains 7 red, 6 white and 4 blue balls. How many selections of three balls on of each colour?

$^{15}C_3 + ^{15}C_{13}$ is equal to:

$^5C_1 + ^5C_2 + ^5C_3 + ^5C_4 + ^5C_5$ is equal to _______.

$^{15}C_{3r} = ^{15}C_{r+3}$, then $r$ is equal to

In an examination a candidate has to pass in 4 of the 4 papers. In how many different ways can be failed?

The Supreme Court Bench consists of 5 judges. In how many ways, the bench can give a majority decision?

There 12 questions to be answered to be Yes or No. How Many ways this can be answered

$^nC_r + ^nC_{r-1} + ^nC_{r-2} + ...$

The number in ways in which 4 persons can occupy 9 vacant seats is

In how many ways can 4 people be selected at random from 6 boys and 4 girls if there are exactly two girls?

Find the number of combinations of the letters of the word COLLEGE taken four together:

A man has 5 friends. In how many ways can be invite one or more of his friends to dinner?

There are 12 points in a plane which are collinear with 4 points, number of triangular that can be formed with the vertices as there points are

The number of diagonals in a polygon of 6 sides

The number of triangles that can be formed by choosing the vertices from a set of 12 points, seven of which lie on the same straight line, is:

Number ways of painting of a face of a cube by 6 colors is

X and Y stand in a line with 6 other people. What is the probability that there are 3 persons between them?

The number of ways of painting the faces of a cube by 6 different colors is

If there are 30 points in a plane of which 5 points are lies on the same line. Then the number of triangles can be formed?

The value $n, r$ if $^nP_r = 3024$ and $^nC_r = 126$

If $^{20}C_x = ^{20}C_{x+6}$, Then the value of $x$ is

A bag contains 4 red, 3 black and 2 white balls > In how many ways 3 balls can be drawn from this bag so that they include at least one black ball?

A Supreme Court Bench consists of 5 judges. In how many ways, the bench can give a majority decision?

The maximum number of points of intersection of $10$ circles will be:

If $^{15}C_r = ^{15}C_{r+3}$, then 'r' is equal to

A Supreme Court Bench consists of $5$ judges. In how many ways, the bench can give a majority division?

In an election, there are five candidates contesting for three vacancies; an elector can vote any number of candidates not exceeding the number of vacancies. In how many ways can one cast his votes?

The no. of ways that $12$ prizes can be divided among $5$ students so that each may give $3$ prizes is

Five balls of different colours are to be placed in three boxes of different sizes. Each box can hold all the five balls. In how many different ways can we place the balls so that no box remains empty?

A box contains $7$ red, $6$ white and $4$ blue balls. How many selections of three balls can be made so that none is red?

In how many ways $3$ prizes out of $5$ can be distributed amongst $3$ brothers equally

There $12$ questions to be answered to be Yes or No. How many ways this can be answered

$^{15}C_r = ^{15}C_{r+3}$, then $r$ is equal to

A polygon has $44$ diagonals then the number of sides are

The number of ways of painting the six faces of a cube with six different given colours is

$^{n}C_1 + ^{n}C_2 + ^{n}C_3 + ............$

A user wants to create a password using $4$ lowercase letters (a-z) and $3$ uppercase letters (A-Z). No letter can be repeated in any form. In how many ways can the password be created if the password must start with an uppercase letter?

A candidate is required to answer $6$ out of $10$ questions, which are divided into two groups each containing $5$ questions and he is not permitted to attempt more than $4$ from each group. In how many ways can he make up his choice?

If $^{n}C_2 = ^{n}C_3$, the value of 'n' is

If $^{2n}C_2 : ^{n}C_2 = 11:1$ the value of $n$ is

There are 12 questions to be answered in Yes or No. How many ways can these be answered?

How many Six-digit telephone numbers can be formed by using 10 distinct digits

The number of ways of arranging 6 boys and 4 girls in a row so that all 4 girls are together is:

$^{15}C_{3r} = ^{15}C_{r+3}$, then '$r$' is equal to

If $^nP_r = 12 ^nP_{r-1}$, then n =

An examination paper consists of 12 questions divided into two parts A and B. Part A contains 7 questions and Part B contains 5 questions. A candidate is required to attempt 8 questions selecting at least 3 from each part, in how many maximum ways can the candidate select the questions?

A boy has 3 library tickets and 8 books of his interest in the library. Out of these 8, he does not want to borrow mathematics part II unless mathematics part-I is also borrowed. In how many ways can he choose the three books to be borrowed?

The number of triangles that can be formed by choosing the vertices from a set of 12 points, Seven of which lie on the same lie on the same straight line is:

A user wants to create a password using 4 lowercase letters (a-z) and 4 uppercase letters (A-Z). No letter can be repeated in any form. In how many ways can the password be created if the password must start with an uppercase letter?