Permutations & Combinations Questions - JEE

Here, $a_r = {}^{2n}C_r$. Find the value of $\displaystyle\prod_{r=1}^{2n}\left(1+\dfrac{a_r}{a_{r-1}}\right)$.

Let $A = \{x_1, x_2, \ldots, x_7\}$ 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 exists exactly three elements $x$ in $A$ such that $f(x) = y_2$, is equal to

(a) If ${}^{22}P_{r+1} : {}^{20}P_{r+2} = 11:52$, find $r$.

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:

The number of four-digit numbers that can be made with the digits 1, 2, 3, 4, and 5 in which at least two digits are identical is

Find the number of positive integers, which can be formed by using any number of digits from 0, 1, 2, 3, 4, 5 but using each digit not more than once in each number. How many of these integers are greater than 3000? What will happen when repetition is allowed?

In how many ways can 10 persons take seats in a row of 24 fixed seats so that no two persons take consecutive seats?

Find the number of ways that 8 beads of different colors be strung as a necklace.

If there are six straight lines in a plane, no two of which are parallel and no three of which pass through the same point, then find the number of points in which these lines intersect.

Find number of non-negative integral solutions of the equation $x + y + z + 2w = 20$.

In how many ways, two different natural numbers can be selected, which are less than or equal to 100 and differ by almost 10?

Twenty guests are to be seated on two sides of a rectangular dinner table, half on each side. Five specific guests have to sit together on any side of the table and another set of four specific guests have to sit on the facing side. Determine the number of sitting arrangements possible.

A class contains three girls and four boys. Every Saturday, five go on a picnic (a different group of students is sent every week). During the picnic, each girl in the group is given a doll by the accompanying teacher. If all possible groups of five have gone for picnic once, the total number of dolls that the girls have got is

How many numbers can be made with the digits 3, 4, 5, 6, 7, 8 lying between 3000 and 4000, which are divisible by 5 while repetition of any digit is not allowed in any number?

Find the number of arrangements of all digits of 12345 such that at least 3 digits will not come in its positions.

We have $\left({}^{21}C_1 + {}^{21}C_2 + \cdots + {}^{21}C_{10}\right)\left({}^{10}C_1 + {}^{10}C_2 + \cdots + {}^{10}C_{10}\right)$Which of the following is the simplified value of the above expression?

If

Five digit numbers are formed using the digits 1, 2, 3, 5, 7 with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1. Then the serial number of 35337 is _____.

If $\binom{n}{9} = \binom{n}{7}$, find $n$.

The letters of the word 'MUMBAI' are arranged in all possible ways as in a dictionary, the rank of the word 'MUMBAI' is

If all permutations of the letters of the word 'AGAIN' are arranged as in a dictionary, then 50th word is

The total number of 4-digit numbers whose greatest common divisor with 54 is 2, is _____.

In how many ways can 12 different balls be divided into three groups of 5, 4 and 3 balls respectively?

A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is

Number of 4-digit numbers (the repetition of digits is allowed) which are made using the digits 1, 2, 3 and 5, and are divisible by 15, is equal to _____

The number of seven digit odd numbers, that can be formed using all the seven digits 1, 2, 2, 2, 3, 3, 5 is _____.

Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11, is equal to _____.

Example 91: In how many ways the sum of upper faces of four distinct dice can be five?

Find the number of ways in which one or more letters can be selected from the letters AAAAA BBBB CCC DD EFG.

If n lines are drawn in a plane such that no two of them are parallel and no three of them are concurrent, such that these lines divide the plane into 67 parts, then find the number of different points at which these lines will cut.

Let 5 digit numbers be constructed using the digits 0, 2, 3, 4, 7, 9 with repetition allowed, and are arranged in ascending order with serial numbers. Then the serial number of the number 42923 is

If $^{2n+1}P_{n-1} : {}^{2n-1}P_n = 11 : 21$, then $n^2 + n + 15$ is equal to:

265. Number of 4 digit numbers of the form $N = abcd$ which satisfy following three conditions:(i) $4000 \leq N (ii) \(N$ is a multiple of 5(iii) \(3 \leq b is equal to:

The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is _____.

The total number of six digit numbers, formed using the digits 4, 5, 9 only and divisible by 6, is _____.

Number of integral solutions to the equation x + y + z = 21, where x $\geq$ 1, y $\geq$ 3, z $\geq$ 4, is equal to _____.

265. Number of 4 digit numbers of the form $N = abcd$ which satisfy following three conditions:(i) $4000 \leq N (ii) \(N$ is a multiple of 5(iii) \(3 \leq b is equal to:

How many five-digit numbers divisible by 6 can be made with the digits 0, 1, 2, 3, 4 and 5 if the digits cannot be repeated in the same number?

The number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining 21 are distinct, is

In how many different ways can a set $A$ of $3n$ elements be partitioned into 3 subsets of equal number of elements? (The subsets $P, Q, R$ form a partition if $P \cup Q \cup R = A$, $P \cap R = \phi$, $Q \cap R = \phi$, $R \cap P = \phi$.)

If the sum of all the 3 digit natural numbers which contain at least one odd digit and one even digit is k. Find k/1000

Four couples (husband and wife) decide to form a committee of four members. The number of different committees that can be formed in which no couple finds a place is $l$, then the sum of digits of $l$ is?

If ${}^nC_3 + {}^nC_4 > {}^{n+1}C_3$, then

Let $T_n$ be the number of all possible triangles formed by joining vertices of an n-sided regular polygon. If $T_{n+1} - T_n = 1$ then the value of n is

Let $ b_n = \sum_{r=0}^{n} \frac{r}{{}^nC_r} $. If $ c_n = \frac{2}{n} $ for all $ n \geq 1 $, find the number of ordered pairs $(p, q)$ of positive integers such that $ 2p^{-1} + 2q^{-1} = 1 $.

m, n are positive integers such that g.c.d. (m, n) = 1 and mn = 25!. The number of rational numbers $\frac{m}{n}$ < 1 is

The number of solutions $(n_1, n_2)$ which satisfy $n_1 n_2 = 2n_1 - n_2$, where $n_1, n_2 \in \mathbb{I}$ (integers) is

In a plane, there are 10 points of which 4 are collinear. How many different straight lines can be drawn joining them?

How many odd numbers of five digits can be formed with the digits 1, 2, 3, 4, 5 if the digits cannot be repeated in the same number?

How many words can be made with the letters of the word BHARATI so that all the vowels are consecutive?

Permutations & Combinations Questions (855)