Permutations & Combinations Questions (855)

How many ways are there to arrange the letters of the word "GARDEN" with the vowels in alphabetical order?

A large pile contains red, white, green and blue balls (all alike except for colour). Find the number of ways to select 20 balls from them such that the selection has at most 3 green balls.

If \(n\) objects are arranged in a row, then the number of ways of selecting three of these objects so that no two of them are next to each other is

In how many ways can 15 members of a council sit along a circular table, when the secretary is to sit on one side of the chairman and the deputy secretary on the other side?

If all the words (with or without meaning) having five letters, formed using the letters of the word SMALL and arranged as in a dictionary; then the position of the word SMALL is

A person predicts the outcome of 20 cricket matches of his home team. Each match can result in either a win, loss, or tie for the home team. Total number of ways in which he can make the predictions so that exactly 10 predictions are correct is equal to

Five balls are to be placed in three boxes. Each can hold all the five balls. In how many different ways can we place the balls so that no box remains empty, if(i) balls and boxes are all different(ii) balls are identical but boxes are different(iii) balls are different but boxes are identical(iv) balls as well as boxes are identical(v) balls as well as boxes are identical but boxes are kept in a row?

Let Tn denote the number of triangles which can be formed using the vertices of a regular polygon of 'n' sides. If Tn+1 − Tn = 21, then 'n' equals:

In how many ways can 30 marks be allotted to 8 questions if each question carries at least 2 marks?

The letters of the word SMALL are arranged as in a dictionary; then the position of the word SMALL is:

In how many ways can 10 people take seats in 24 fixed seats so that out of every pair of seats equidistant from the beginning and end at least one seat is empty?

In a certain algebraical exercise book there are 4 examples on arithmetical progressions, 5 examples on permutation and combination, and 6 examples on binomial theorem. Find the number of ways a teacher can select for his pupils at least one but not more than 2 examples from each of these sets.

Find the sum of all the numbers that can be formed with the digits 2, 3, 4, 5 taken all at a time.

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 natural numbers, between $212$ and $999$, such that the sum of their digits is $15$, is \rule{2cm}{0.4pt}.

Find the sum of all the numbers greater than 10,000 that can be made with the digits 1, 3, 5, 7 and 9 if digits are not repeated in the same number.

Number of natural numbers with 5 distinct digits formed using $\{1,2,3,5,6,7,9\}$ and divisible by 6 are

The lines $L_1,L_2,\ldots,L_{20}$ are distinct. For $n=1,2,3,\ldots,10$ all the lines $L_{2n-1}$ are parallel to each other and all the lines $L_{2n}$ pass through a given point $P$. The maximum number of points of intersection of pairs of lines from the set $\{L_1,L_2,\ldots,L_{20}\}$ is equal to:

Twelve different letters are to be put in twelve pockets in a row. If five of the pockets are too small for six of the letters then in how many different ways can the letters be put in the pockets?

5 girls and 7 boys seated at a round table so no two girls sit together. Number of ways is

How many numbers of six digits can be made by arranging the digits of the number 123425 so that all the even digits do not occupy consecutive places?

Number of words of 4 letters that can be formed with the letters of the word IIT JEE, is

Find the number of ways in which the birthdays of six different persons will fall in exactly two calendar months.

There are 16 points in a plane of which 8 are on one line and 8 are on another line. The number of triangles that can be formed with the points as vertices is ______.

When P is a natural number, then \(\frac{P^{n+1}-(P-1)^{2n-1}}{P}\) is divisible by

How many integers > 100 and < 106 have the digital sum = 5?

The number of three-digit numbers $\overline{abc}$ which satisfy $a \leq b > c$ is

In an examination of nine papers, a candidate has to pass in more papers than the number of papers in which he fails in order to be successful. The number of ways in which he can be unsuccessful is

How many five digit numbers can be formed from 1, 2, 3, 4, 5 (without repetition), when the digit at the unit place must be greater than that in the tenth place?

There are unlimited number of identical balls of four different colours. How many arrangements of at most 8 balls in a row can be made by using them?

Seven different coins are to be divided amongst three persons. If no two of the persons receive the same number of coins but each receives atleast one coin & none is left over, then the number of ways it which the division may be made is:

The number of $6$-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is \rule{2cm}{0.4pt}.

The number of ordered triplets (x, y, z) of non-negative integers satisfying the conditions x + y + z \leq 100 and x \leq y \leq zIf x is odd, the count is

Number of 7-digit positive integers using only $\{1,2,3,4\}$ with digit sum $=12$ is_______.

There are 10 points on a plane of which 5 points are collinear. Also, no three of the remaining 5 points are collinear. Then find:(i) the number of straight lines joining these points.(ii) the number of triangles formed by joining these points.

There are an even number of points in a plane of which half are lying on the same line and no other three are collinear. If the total number of triangles that can be made with vertices at these points is 110, then the number of given points is

Let $P$ be the set of seven-digit numbers with sum of their digits equal to $11$. If the numbers in $P$ are formed by using the digits $1,2$ and $3$ only, then the number of elements in $P$ is:

How many 3-digit numbers can be formed using the digits \(0, 1, 2, 2, 3, 3, 3\)? (Repetition according to availability only)

There is a rectangular sheet of dimension \((2m - 1) \times (2n - 1)\), (where \(m > 0, n > 0\)). It has been divided into squares of unit area by drawing lines perpendicular to the sides. Find the number of rectangles having sides of odd unit length.

8 clay targets are arranged as shown. If \(N\) be the number of different ways they can be shot (one at a time) if no target can be shot until the target(s) below it have been shot. Find the ten's digit of \(N\).

The total number of ways in which 15 identical blankets can be distributed among four persons so that each of them gets at least two blankets is equal to

12 boys and 2 girls are to be seated in a row such that there are atleast 3 boys between the 2 girls. The number of ways this can be done is λ × 12!. Find the value of λ.

Two players \(P_1\) and \(P_2\) play a series of \(2n\) games. Each game can result in either a win or a loss for \(P_1\). The total number of ways in which \(P_1\) can win the series of these games is equal to

If the letters of words SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number

Find the number of ordered pairs \((x, y)\) if \(x, y \in \{0, 1, 2, 3, \ldots, 10\}\) and if \(|x - y| > 5\).

If \(x

If \({}^n P_r = {}^n P_{r+1}\) and \({}^n C_r = {}^n C_{r-1}\), then the value of \(n + r\) is ___.

Find the number of ways in which 16 constables can be assigned to patrol 8 villages, 2 for each.

The number of different $5$-digit numbers greater than $50000$ that can be formed using the digits $0,1,2,3,4,5,6,7$, such that the sum of their first and last digits should not be more than $8$, is:

Total 3-digit numbers divisible by 3 from $\{1,3,5,8\}$ with repetition is