Permutations & Combinations · JEE Main previous year questions

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 family members are not separated?

1. A.$2!3!4!$
2. B.$(3!{)}^3\cdot (4!)$
3. C.$3!(4!{)}^3$
4. D.$(3!{)}^2\cdot (4!)$

The position of a moving car at time t is given by $f(t)=a{t}^2+bt+c$, $t>0$, where $a,b$ and $c$ are real numbers greater than 1. Then the average speed of the car over the time interval $[t_1,t_2]$ is attained at the point :

1. A.$\frac{(t_1+t_2)}{2}$
2. B.$\frac{(t_2-t_1)}{2}$
3. C.$2a(t_1+t_2)+b$
4. D.$a(t_2-t_1)+b$

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 ________ .

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 of the word 'MOTHER' is ______.

For a positive integer $n$, ${\left(1+\frac{1}{x}\right)}^n$ is expanded in increasing powers of $x$. If three consecutive coefficients in this expansion are in the ratio, $2:5:12$, then $n$ is equal to _______.

If the number of five digit numbers with distinct digits and 2 at the $10^{\mathrm{th}}$ place is $336\mathrm{k}$, then $\mathrm{k}$ is equal to :

1. A.6
2. B.8
3. C.4
4. D.7

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 of them are red is ___________.

The number of 4 letter words (with or without meaning) that can be formed from the eleven letters of the word 'EXAMINATION' is _______.

Total number of 6-digit numbers in which only and all the five digits 1, 3, 5, 7 and 9 appear, is :

1. A.$\frac{5}{2}(6!)$
2. B.$6!$
3. C.$5^6$
4. D.$\frac{1}{2}(6!)$

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

1. A.$2^{20}-1$
2. B.$2^{20}$
3. C.$2^{20}+1$
4. D.$2^{21}$

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 such that there is at least one boy and at least one girl in each team, is 1750, then n is equal to :

1. A.24
2. B.25
3. C.27
4. D.28

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 :

1. A.36
2. B.60
3. C.72
4. D.48

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 is the number of ways the committee is formed with at least 3 females, then :

1. A.n = m – 8
2. B.m = n = 78
3. C.m + n = 68
4. D.m = n = 68

Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are addded to the total number of balls used in forming the equilaterial triangle, then all these balls can be arranged in a square whose each side contains exactly 2 balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is :-

1. A.262
2. B.190
3. C.157
4. D.225

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 :

1. A.306
2. B.288
3. C.310
4. D.360

Consider three boxes, each containing, 10 balls labelled 1, 2, ..., 10. Suppose one ball is randomly drawn from each of the boxes. Denote by $n_i$, the label of the ball drawn from the $i^{\text{th}}$ box, $(i=1,2,3)$. Then, the number of ways in which the balls can be chosen such that $n_1<n_2<n_3$ is :

1. A.164
2. B.240
3. C.82
4. D.120

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 of 3 is :

1. A.24
2. B.30
3. C.36
4. D.48

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 :

1. A.6
2. B.7
3. C.8
4. D.9

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 dictionary is always in the middle. The number of such arrangements is :

1. A.at least 750 but less than 1000
2. B.at least 1000
3. C.less than 500
4. D.at least 500 but less than 750

The number of ways in which 5 boys and 3 girls can be seated on a round table if a particular boy ${\mathrm{B}}_1$ and a particular girl ${\mathrm{G}}_1$ never sit adjacent to each other, is :

1. A.$5\times 6!$
2. B.$6\times 6!$
3. C.$7!$
4. D.$5\times 7!$

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 :

1. A.$44^{\mathrm{th}}$
2. B.$45^{\mathrm{th}}$
3. C.$46^{\mathrm{th}}$
4. D.$47^{\mathrm{th}}$

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 :

1. A.$46^{th}$
2. B.$59^{th}$
3. C.$52^{nd}$
4. D.$58^{th}$

The number of integers greater than 6,000 that can be formed, using the digits 3, 5, 6, 7 and 8, without repetition, is:

1. A.120
2. B.72
3. C.216
4. D.192

If seven women and seven men are to be seated around a circular table such that there is a man on either side of every woman, then the number of seating arrangements is

1. A.$6!7!$
2. B.$(6!{)}^2$
3. C.$(7!{)}^2$
4. D.$7!$