JEE Main
Mathematics
Permutations and Combinations
Previous Years Questions

## Numerical

Number of integral solutions to the equation $$x+y+z=21$$, where $$x \ge 1,y\ge3,z\ge4$$, is equal to ____________.
The total number of six digit numbers, formed using the digits 4, 5, 9 only and divisible by 6, is ____________.
The number of 3-digit numbers, that are divisible by either 2 or 3 but not divisible by 7, is ____________.
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...
Let $\mathrm{A}=\left[\mathrm{a}_{i j}\right], \mathrm{a}_{i j} \in \mathbb{Z} \cap[0,4], 1 \leq i, j \leq 2$. The number of matrices A such that the ...
If ${ }^{2 n+1} \mathrm{P}_{n-1}:{ }^{2 n-1} \mathrm{P}_{n}=11: 21$, then $n^{2}+n+15$ is equal to :
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. ...
Number of 4-digit numbers that are less than or equal to 2800 and either divisible by 3 or by 11 , is equal to ____________.
The number of seven digits odd numbers, that can be formed using all theseven digits 1, 2, 2, 2, 3, 3, 5 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 total number of 4-digit numbers whose greatest common divisor with 54 is 2, is __________.
If all the six digit numbers $$x_1\,x_2\,x_3\,x_4\,x_5\,x_6$$ with $$0... Five digit numbers are formed using the digits 1, 2, 3, 5, 7 with repetitions and are written in descending order with serial numbers. For example, th... A triangle is formed by X-axis, Y-axis and the line$$3x+4y=60$$. Then the number of points P(a, b) which lie strictly inside the triangle, where a is... Suppose Anil's mother wants to give 5 whole fruits to Anil from a basket of 7 red apples, 5 white apples and 8 oranges. If in the selected 5 fruits, a... Let$$x$$and$$y$$be distinct integers where$$1 \le x \le 25$$and$$1 \le y \le 25$$. Then, the number of ways of choosing$$x$$and$$y$$, such t... A boy needs to select five courses from 12 available courses, out of which 5 courses are language courses. If he can choose at most two language cours... The number of 9 digit numbers, that can be formed using all the digits of the number 123412341 so that the even digits occupy only even places, is ___... The number of natural numbers lying between 1012 and 23421 that can be formed using the digits$$2,3,4,5,6$$(repetition of digits is not allowed) and... A class contains b boys and g girls. If the number of ways of selecting 3 boys and 2 girls from the class is 168 , then$$\mathrm{b}+3 \mathrm{~g}$$i... Let$$S$$be the set of all passwords which are six to eight characters long, where each character is either an alphabet from$$\{A, B, C, D, E\}$$or... Numbers are to be formed between 1000 and 3000 , which are divisible by 4 , using the digits$$1,2,3,4,5$$and 6 without repetition of digits. Then th... The number of 5-digit natural numbers, such that the product of their digits is 36 , is __________. The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number ... The number of 6-digit numbers made by using the digits 1, 2, 3, 4, 5, 6, 7, without repetition and which are multiple of 15 is ____________. The total number of four digit numbers such that each of first three digits is divisible by the last digit, is equal to ________. Let b1b2b3b4 be a 4-element permutation with bi$$\in$${1, 2, 3, ........, 100} for 1$$\le$$i$$\le$$4 and bi$$\ne$$bj for i$$\ne$$j, such tha... The number of ways, 16 identical cubes, of which 11 are blue and rest are red, can be placed in a row so that between any two red cubes there should b... The total number of 3-digit numbers, whose greatest common divisor with 36 is 2, is ___________. There are ten boys B1, B2, ......., B10 and five girls G1, G2, ........, G5 in a class. Then the number of ways of forming a group consisting of three... The total number of three-digit numbers, with one digit repeated exactly two times, is ______________. The number of 3-digit odd numbers, whose sum of digits is a multiple of 7, is _____________. The number of 7-digit numbers which are multiples of 11 and are formed using all the digits 1, 2, 3, 4, 5, 7 and 9 is __________. In an examination, there are 5 multiple choice questions with 3 choices, out of which exactly one is correct. There are 3 marks for each correct answe... All the arrangements, with or without meaning, of the word FARMER are written excluding any word that has two R appearing together. The arrangements a... The number of six letter words (with or without meaning), formed using all the letters of the word 'VOWELS', so that all the consonants never come tog... A number is called a palindrome if it reads the same backward as well as forward. For example 285582 is a six digit palindrome. The number of six digi... If$${}^1{P_1} + 2.{}^2{P_2} + 3.{}^3{P_3} + .... + 15.{}^{15}{P_{15}} = {}^q{P_r} - s,0 \le s \le 1$$, then$${}^{q + s}{C_{r - s}}$$is equal to ___... The number of three-digit even numbers, formed by the digits 0, 1, 3, 4, 6, 7 if the repetition of digits is not allowed, is ______________. Let n be a non-negative integer. Then the number of divisors of the form "4n + 1" of the number (10)10 . (11)11 . (13)13 is equal to ___________.... There are 5 students in class 10, 6 students in class 11 and 8 students in class 12. If the number of ways, in which 10 students can be selected from ... If the digits are not allowed to repeat in any number formed by using the digits 0, 2, 4, 6, 8, then the number of all numbers greater than 10,000 is ... There are 15 players in a cricket team, out of which 6 are bowlers, 7 are batsman and 2 are wicketkeepers. The number of ways, a team of 11 players be... If$$\sum\limits_{r = 1}^{10} {r!({r^3} + 6{r^2} + 2r + 5) = \alpha (11!)} $$, then the value of$$\alpha$$is equal to ___________. The number of times the digit 3 will be written when listing the integers from 1 to 1000 is The missing value in the following figure is ... The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1, 2, 3, 4, 5, if the repetition of digits is not allowed a... The students S1, S2, ....., S10 are to be divided into 3 groups A, B and C such that each group has at least one student and the group C has at most 3... The sum of first four terms of a geometric progression (G. P.) is$${{65} \over {12}}$$and the sum of their respective reciprocals is$${{65} \over {...
The number of words (with or without meaning) that can be formed from all the letters of the word “LETTER” in which vowels never come together is ____...
Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up a three or a five, is _________.
The number of words, with or without meaning, that can be formed by taking 4 letters at a time from the letters of the word ’SYLLABUS’ such that two l...
A test consists of 6 multiple choice questions, each having 4 alternative answers of which only one is correct. The number of ways, in which a candida...
The total number of 3-digit numbers, whose sum of digits is 10, is ______.
If the letters of the word 'MOTHER' be permuted and all the words so formed (with or without meaning) be listed as in a dictionary, then the position ...
The number of 4 letter words (with or without meaning) that can be formed from the eleven letters of the word 'EXAMINATION' is _______.
An urn contains 5 red marbles, 4 black marbles and 3 white marbles. Then the number of ways in which 4 marbles can be drawn so that at the most three ...

The value of $$\frac{1}{1 ! 50 !}+\frac{1}{3 ! 48 !}+\frac{1}{5 ! 46 !}+\ldots .+\frac{1}{49 ! 2 !}+\frac{1}{51 ! 1 !}$$ is :
The number of ways of selecting two numbers $a$ and $b, a \in\{2,4,6, \ldots ., 100\}$ and $b \in\{1,3,5, \ldots . ., 99\}$ such that 2 is the remaind...
The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of...
The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48, is
The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1, 3, 5, 7, 9 without repetition, is
$$\sum\limits_{k = 0}^6 {{}^{51 - k}{C_3}}$$ is equal to
The number of integers, greater than 7000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition is
The number of ways to distribute 30 identical candies among four children C1, C2, C3 and C4 so that C2 receives at least 4 and at most 7 candies, C3 r...
The total number of 5-digit numbers, formed by using the digits 1, 2, 3, 5, 6, 7 without repetition, which are multiple of 6, is
Let P1, P2, ......, P15 be 15 points on a circle. The number of distinct triangles formed by points Pi, Pj, Pk such that i +j + k $$\ne$$ 15, is : ...
If $${}^n{P_r} = {}^n{P_{r + 1}}$$ and $${}^n{C_r} = {}^n{C_{r - 1}}$$, then the value of r is equal to :
The sum of all the 4-digit distinct numbers that can be formed with the digits 1, 2, 2 and 3 is :
If the sides AB, BC and CA of a triangle ABC have 3, 5 and 6 interior points respectively, then the total number of triangles that can be constructed ...
Team 'A' consists of 7 boys and n girls and Team 'B' has 4 boys and 6 girls. If a total of 52 single matches can be arranged between these two teams w...
Consider a rectangle ABCD having 5, 7, 6, 9 points in the interior of the line segments AB, CD, BC, DA respectively. Let $$\alpha$$ be the number of t...
A natural number has prime factorization given by n = 2x3y5z, where y and z are such that y + z = 5 and y$$-$$1 + z$$-$$1 = $${5 \over 6}$$, y > z....
The number of seven digit integers with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only is :
The total number of positive integral solutions (x, y, z) such that xyz = 24 is :
A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as India...
Two families with three members each and one family with four members are to be seated in a row. In how many ways can they be seated so that the same ...
There are 3 sections in a question paper and each section contains 5 questions. A candidate has to answer a total of 5 questions, choosing at least on...
The value of (2.1P0 – 3.2P1 + 4.3P2 .... up to 51th term) + (1! – 2! + 3! – ..... up to 51th term) is equal to :...
Let n > 2 be an integer. Suppose that there are n Metro stations in a city located along a circular path. Each pair of stations is connected by a s...
If the number of five digit numbers with distinct digits and 2 at the 10th place is 336 k, then k is equal to :
If a, b and c are the greatest value of 19Cp, 20Cq and 21Cr respectively, then
The number of ordered pairs (r, k) for which 6.35Cr = (k2 - 3). 36Cr + 1, where k is an integer, is: ...
Total number of 6-digit numbers in which only and all the five digits 1, 3, 5, 7 and 9 appear, is:
A group of students comprises of 5 boys and n girls. If the number of ways, in which a team of 3 students can randomly be selected from this group suc...
The number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining 21 are distinct, is :
Suppose that 20 pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar has been connected by...
The number of 6 digit numbers that can be formed using the digits 0, 1, 2, 5, 7 and 9 which are divisible by 11 and no digit is repeated is :
A committee of 11 members is to be formed from 8 males and 5 females. If m is the number of ways the committee is formed with at least 6 males and n i...
The number of four-digit numbers strictly greater than 4321 that can be formed using the digits 0,1,2,3,4,5 (repetition of digits is allowed) is :
All possible numbers are formed using the digits 1, 1, 2, 2, 2, 2, 3, 4, 4 taken all at a time. The number of such numbers in which the odd digits occ...
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 gam...
Consider three boxes, each containing, 10 balls labelled 1, 2, … , 10. Suppose one ball is randomly drawn from each of the boxes. Denote by ni, the la...
If  $$\sum\limits_{r = 0}^{25} {\left\{ {{}^{50}{C_r}.{}^{50 - r}{C_{25 - r}}} \right\} = K\left( {^{50}{C_{25}}} \right)} ,\,\,$$ then K is...
Let S be the set of all triangles in the xy-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integra...
The number of natural numbers less than 7,000 which can be formed by using the digits 0, 1, 3, 7, 9 (repitition of digits allowed) is equal to :
Consider a class of 5 girls and 7 boys. The number of different teams consisting of 2 girls and 3 boys that can be formed from this class, if there ar...
The number of numbers between 2,000 and 5,000 that can be formed with the digits 0, 1, 2, 3, 4 (repetition of digits is not allowed) and are multiple ...
From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictio...
The number of four letter words that can be formed using the letters of the word BARRACK is :
n$$-$$digit numbers are formed using only three digits 2, 5 and 7. The smallest value of n for which 900 such distinct numbers can be formed, is :
The number of ways in which 5 boys and 3 girls can be seated on a round table if a particular boy B1 and a particular girl G1 never sit adjacent to e...
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 positio...
A man X has 7 friends, 4 of them are ladies and 3 are men. His wife Y also has 7 friends, 3 of them are ladies and 4 are men. Assume X and Y have no c...
If    $${{{}^{n + 2}C{}_6} \over {{}^{n - 2}{P_2}}}$$ = 11, then n satisfies the equation :
The sum $$\sum\limits_{r = 1}^{10} {\left( {{r^2} + 1} \right) \times \left( {r!} \right)}$$ is equal to :
If the four letter words (need not be meaningful ) are to be formed using the letters from the word “MEDITERRANEAN” such that the first letter is R an...
The value of $$\sum\limits_{r = 1}^{15} {{r^2}} \left( {{{{}^{15}{C_r}} \over {{}^{15}{C_{r - 1}}}}} \right)$$ is equal to :
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 po...
The number of integers greater than 6,000 that can be formed, using the digits 3, 5, 6, 7 and 8, without repetition, is:
Let $${T_n}$$ be the number of all possible triangles formed by joining vertices of an n-sided regular polygon. If $${T_{n + 1}} - {T_n}$$ = 10, then ...
Let A and B be two sets containing 2 elements and 4 elements respectively. The number of subsets of A $$\times$$ B having 3 or more elements is -
Assuming the balls to be identical except for difference in colours, the number of ways in which one or more balls can be selected from 10 white, 9 gr...
These are 10 points in a plane, out of these 6 are collinear, if N is the number of triangles formed by joining these points. then:
Statement - 1: The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is emply is $${}^9{C_3}$$. Statement -...
There are two urns. Urn A has 3 distinct red balls and urn B has 9 distinct blue balls. From each urn two balls are taken out at random and then trans...
From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictio...
In a shop there are five types of ice-cream available. A child buys six ice-cream. Statement - 1: The number of different ways the child can buy th...
How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent?
The set S = {1, 2, 3, ........., 12} is to be partitioned into three sets A, B, C of equal size. Thus $$A \cup B \cup C = S,\,A \cap B = B \cap C = A ... At an election, a voter may vote for any number of candidates, not greater than the number to be elected. There are 10 candidates and 4 are of be sele... If the letter of the word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at s... How many ways are there to arrange the letters in the word GARDEN with vowels in alphabetical order The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is The number of ways in which 6 men and 5 women can dine at a round table if no two women are to sit together is given by A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choic... If$${}^n{C_r}$$denotes the number of combination of n things taken r at a time, then the expression$$\,{}^n{C_{r + 1}} + {}^n{C_{r - 1}} + 2\, \tim...
Total number of four digit odd numbers that can be formed using 0, 1, 2, 3, 5, 7 (using repetition allowed) are
Number greater than 1000 but less than 4000 is formed using the digits 0, 1, 2, 3, 4 (repetition allowed). Their number is
The sum of integers from 1 to 100 that are divisible by 2 or 5 is
Five digit number divisible by 3 is formed using 0, 1, 2, 3, 4 and 5 without repetition. Total number of such numbers are
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