Chapter 7

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? [Sep. 06, 2020 (I)] (a) \(2 ! 3 ! 4 !\) (b) \((3 !)^{3} \cdot(4 !)\) (c) \((3 !)^{2},(4 !)\) (d) \(3 !(4 !)^{3}\)

The value of \(\left(2 \cdot{ }^{1} P_{0}-3 \cdot{ }^{2} P_{1}+4 \cdot{ }^{3} P_{2}-\ldots\right.\) up to \(51^{\text {th }}\) term \()\) \(+\left(1 !-2 !+3 !-\ldots\right.\) up to \(51^{\text {th }}\) term ) is equal to : \([\) Sep. \(03,2020(\mathrm{I})]\) (a) \(1-51(51) !\) (b) \(1+(51) !\) (c) \(1+(52) !\) (d) 1

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

If the number of five digit numbers with distinct digits and 2 at the \(10^{\text {a }}\) place is \(336 \mathrm{k}\), then \(\mathrm{k}\) is equal to: (a) 4 (b) 6 (c) 7 (d) 8

Total number of 6 -digit numbers in which only and all the five digits \(1,3,5,7\) and 9 appear, is: (a) \(\frac{1}{2}(6 !)\) (b) 6 (c) \(5^{6}\) (d) \(\frac{5}{2}(6 !)\)

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: [April 08, 2019(II)] (a) 288 (b) 360 (c) 306 (d) 310

Consider three boxes, each containing 10 balls labelled \(1,2, \ldots, 10 .\) Suppose one ball is randomly drawn from each of the boxes. Denote byn \(_{i}\), the label of the ball drawn from the \(l^{\text {th }}\) box, \((i=1,2,3)\). Then, the number of ways in which the balls can be chosen such that \(\mathrm{n}_{1}<\mathrm{n}_{2}<\mathrm{n}_{3}\) is : (a) 120 (b) 82 (c) 240 (d) 164

The number of natural numbers less than 7,000 which can be formed by using the digits \(0,1,3,7,9\) (repetition of digits allowed) is equal to; (a) 374 (b) 372 (c) 375 (d) 250

Let \(\mathrm{S}\) be the set of all triangles in the \(\mathrm{xy}\)-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in \(\mathrm{S}\) has area \(50 \mathrm{sq}\). units, then the number of elements in the set \(\mathrm{S}\) is: (a) 9 (b) 18 (c) 36 (d) 32

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? (a) 30 (b) 48 (c) 24 (d) 36

\(n\) - 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 (a) 6 (b) 8 (c) 9 (d) 7

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 : (a) \(5 \times 6 !\) (b) \(6 \times 6 !\) (c) 7 (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: (a) \(44^{\text {th }}\) (b) \(45^{\text {th }}\) (c) \(46^{\text {th }}\) (d) \(47^{\text {th }}\)

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: (a) \(52^{\text {nd }}\) (b) \(58^{\text {th }}\) (c) \(46^{\text {th }}\) (d) \(59^{\text {th }}\)

The sum \(\sum_{\mathrm{r}=1}^{10}\left(\mathrm{r}^{2}+1\right) \times(\mathrm{r} !)\) is equal to : (a) \(11 \times(11 !)\) (b) \(10 \times(11 !)\) (c) (11!) (d) \(101 \times(10 !)\)

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 \(\mathrm{R}\) and the fourth letter is \(\mathrm{E}\), then the total number of all such words is : (a) 110 (b) 59 (c) \(\frac{11 !}{(2 !)^{3}}\) (d) 56

The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices \((0,0),(0,41)\) and \((41,0)\) is: (a) 820 (b) 780 (c) 901 (d) 861

The number of integers greater than 6,000 that can be formed, using the digits \(3,5,6,7\) and 8, without repetition, is: (a) 120 (b) 72 (c) 216 (d) 192

The number of ways of selecting 15 teams from 15 men and 15 women, such that each team consists of a man and a woman, is: (a) 1120 (b) 1880 (c) 1960 (d) 1240

Two women and some men participated in a chess tournament in which every participant played two games with each of the other participants. If the number of games that the men played between themselves exceeds the number of games that the men played with the women by 66 , then the number of men who participated in the tournament lies in the interval: (a) \([8,9]\) (b) \([10,12)\) (c) \((11,13]\) (d) \((14,17)\)

The sum of the digits in the unit's place of all the 4 -digit numbers formed byusing the numbers \(3,4,5\) and 6, without repetition, is: \(\quad\) Online April 9, 2014] (a) 432 (b) 108 (c) 36 (d) 18

5 - digit numbers are to be formed using \(2,3,5,7,9\) without repeating the digits. If \(p\) be the number of such numbers that exceed 20000 and \(q\) be the number of those that lie between 30000 and 90000 , then \(p: q\) is: (a) \(6: 5\) (b) \(3: 2\) (c) \(4: 3\) (d) \(5: 3\)

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 green and 7 black balls is: (a) 880 (b) 629 (c) 630 (d) 879

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 (a) \(6 ! 7 !\) (b) \((6 !)^{2}\) (c) \((7 !)^{2}\) (d) \(7 !\)

If the letters 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 serial number [2005] (a) 601 (b) 600 (c) 603 (d) 602

How many ways are there to arrange the letters in the word GARDEN with vowels in alphabetical order[2004] (a) 480 (b) 240 (c) 360 (d) 120

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) \(6 ! \times 5 !\) (b) \(6 \times 5\) (c) 30 (d) \(5 \times 4\)

The sum of integers from 1 to 100 that are divisible by 2 or 5 is (a) 3000 (b) 3050 (c) 3600 (d) 3250

Number greater than 1000 but less than 4000 is formed using the digits \(0,1,2,3,4\) (repetition allowed). Their number is (a) 125 (b) 105 (c) 374 (d) 625

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

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 one question from each section. Then the number of ways, in which the candidate can choose the questions, is: \(\quad\) [Sep. 05, 2020 (II)] (a) 3000 (b) 1500 (c) 2255 (d) 2250

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 candidate answers all six questions such that exactly four of the answers are correct, is _______.

The total number of 3 -digit numbers, whose sum of digits is \(10 .\) is

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 straight track only. Further, each pair of nearest stations is connected by blue line, whereas all remaining pairs of stations are connected by red line. If thenumber of red lines is 99 times the number of blue lines, then the value of \(n\) is: (a) 201 (b) 200 (c) 101 (d) 199

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

If \(a, b\) and \(c\) are the greatest values of \({ }^{19} C_{p},{ }^{20} C_{g}\) and \({ }^{21} C\). respectively, then: (a) \(\frac{a}{11}=\frac{b}{22}=\frac{c}{21}\) (b) \(\frac{a}{10}=\frac{b}{11}=\frac{c}{21}\) (c) \(\frac{a}{11}=\frac{b}{22}=\frac{c}{42}\) (d) \(\frac{a}{10}=\frac{b}{11}=\frac{c}{42}\)

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

The number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining 21 are distinct is: (a) \(2^{20}-1\) (b) \(2^{23}\) (c) \(2^{20}\) (d) \(2^{x}+1\)

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: (a) 28 (b) 27 (c) 25 (d) 24

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 beams with the top of all its non-adjacent pillars, then the total number of beams is : (a) 170 (b) 180 (c) 210 (d) 190

A committee of 11 members is to be formed from 8 males and 5 females. If \(\mathrm{m}\) 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: (a) \(\mathrm{m}+\mathrm{n}=68\) (b) \(\mathrm{m}=\mathrm{n}=78\) (c) \(\mathrm{n}=\mathrm{m}-8\) (d) \(\mathrm{m}=\mathrm{n}=68\)

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 occupy even places is : (a) 180 (b) 175 (c) 160 (d) 162

If \(\sum_{i=1}^{20}\left(\frac{{ }^{20} \mathrm{C}_{i-1}}{{ }^{20} \mathrm{C}_{i}+{ }^{20} \mathrm{C}_{i-1}}\right)^{3}=\frac{k}{21}\), then \(k\) equals: (a) 400 (b) 50 (c) 200 (d) 100

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 are two specific boys A and \(\mathrm{B}\), who refuse to be the members of the same team, is: (a) 500 (b) 200 (c) 300 (d) 350

The number of four letter words that can be formed using the letters of the word BARRACK is (a) 144 (b) 120 (c) 264 (d) 270

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: (a) less than 500 (b) at least 500 but less than 750 (c) at least 750 but less than 1000 (d) at least 1000

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 \(\mathrm{X}\) and \(\mathrm{Y}\) have no common friends. Then the total number of ways in which \(\mathrm{X}\) and Y together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of \(\mathrm{X}\) and \(\mathrm{Y}\) are in this party, is : \([2017 \mid\) (a) 484 (b) 485 (c) 468 (d) 469

If \(\frac{{ }^{\mathrm{n}+2} \mathrm{C}_{6}}{\mathrm{n}-2} \mathrm{P}_{2}=11\), then \(\mathrm{n}\) satisfies the equation : (a) \(\mathrm{n}^{2}+\mathrm{n}-110=0\) (b) \(\mathrm{n}^{2}+2 \mathrm{n}-80=0\) (c) \(\mathrm{n}^{2}+3 \mathrm{n}-108=0\) (d) \(\mathrm{n}^{2}+5 \mathrm{n}-84=0\)

The value of \(\sum_{\mathrm{r}=1}^{15} \mathrm{r}^{2}\left(\frac{{ }^{15} \mathrm{C}_{\mathrm{r}}}{{ }^{15} \mathrm{C}_{\mathrm{r}-1}}\right)\) is equal to : (a) 1240 (b) 560 (c) 1085 (d) 680

Let \(\mathrm{A}\) and \(\mathrm{B}\) be two sets containing four and two elements respectively. Then the number of subsets of the set \(\mathrm{A} \times \mathrm{B}\), each having at least three elements is: (a) 275 (b) 510 (c) 219 (d) 256