Chapter 7

Let \(y\) be an element of the set \(A=\\{1,2,3,5,6,10,15\), \(30\\}\) and \(x_{1}, x_{2}, x_{3}\) be integers such that \(x_{1} x_{2} x_{3}=y\), then the number of positive integral solutions of \(x_{1} x_{2} x_{3}=y\) is (A) 64 (B) 27 (C) 81 (D) None of these

If \(m=\) number of distinct rational numbers \(\frac{p}{q} \in\) \((0,1)\) such that \(p, q \in\\{1,2,3,4,5\\}\) and \(n=\) number of mappings from \(\\{1,2,3\\}\) onto \(\\{1,2\\}\), then \(m-n\) is (A) 1 (B) \(-1\) (C) 0 (D) None of these

The letters of the word RANDOM are written in all possible orders and these words are written out as in a dictionary then the rank of the word RANDOM is (A) 614 (B) 615 (C) 613 (D) 616

If eight persons are to address a meeting then the number of ways in which a specified speaker is to speak before another specified speaker, is (A) 40320 (B) 2520 (C) 20160 (D) None of these

The number of permutations of letters \(a, b, c, d, e, f, g\) so that neither the pattern beg nor cad appears is (A) \(\frac{7 !}{3 ! 3 !}\) (B) \(\frac{7 !}{2 ! 3 ! 3 !}\) (C) 4806 (D) None of these

The number of ways of selecting 10 balls from the unlimited number of red, green, white and yellow balls, if selection must include 2 red and 3 yellow balls, is (A) 36 (B) 56 (C) 112 (D) None of these

Let \(A=\\{1,2,3,4\\}\) and \(B=\\{1,2\\}\). Then, the number of onto functions from \(A\) to \(B\) is: (A) 8 (B) 14 (C) 12 (D) None of these

Given five line segments of lengths \(2,3,4,5,6\) units. Then the number of triangles that can be formed by joining these lines is (A) \({ }^{5} C_{3}-3\) (B) \({ }^{5} C_{3}-1\) (C) \({ }^{5} C_{3}\) (D) \({ }^{5} C_{3}-2\)

If \(a\) represents the number of permutations of \((x+2)\) things taken together, \(b\) represents the number of permutations of 11 things taken together out of \(x\) things, and \(c\) represents the number of permutations of \((x-11)\) things taken together so that \(a=182 b c\), then \(x=\) (A) 15 (B) 12 (C) 10 (D) 18

How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even positions? (A) 16 (B) 36 (C) 60 (D) 180

For a game in which two partners play against two other partners, six persons are available. If every possible pair must play with every other possible pair, then the total number of games played is (A) 90 (B) 45 (C) 30 (D) 60

A five digit number divisible by 3 is to be formed using the numerals \(0,1,2,3,4\) and 5 without repetition. The total number of ways this can be done is (A) 216 (B) 600 (C) 240 (D) 3125

A box contains two white balls, three black balls and four red balls. The number of ways in which three balls can be drawn from the box if atleast one black ball is to be included in the draw, is (A) 32 (B) 64 (C) 128 (D) None of these

The sum of all the numbers that can be formed with the digits \(2,3,4,5\) taken all at a time is (A) 66666 (B) 84844 (C) 93324 (D) None of these

If the number of ways in which \(n\) different things can be distributed among \(n\) persons so that at least one person does not get any thing is 232 . Then \(n\) is equal to (A) 3 (B) 4 (C) 5 (D) None of these

Every body in a room shakes hands with every body else. The total number of hand shakes is \(66 .\) The total number of persons in the room is (A) 11 (B) 12 (C) 13 (D) 14

\({ }^{m} C_{r+1}+=\sum_{k=m}^{n}{\underline{\phantom{xx}}}^{k} C_{r}=\) (A) \({ }^{n} C_{r+1}\) (B) \({ }^{n+1} C_{r+1}\) (C) \({ }^{n} C_{r}\) (D) None of these

Two straight lines intersect at a point \(O\). Points \(A_{1}\), \(A_{2}, \ldots, A_{n}\) are taken on one line and points \(B_{1}, B_{2}, \ldots\) \(B_{n}\) on the other. If the point \(O\) is not to be used, the number of triangles that can be drawn using these points as vertices, is (A) \(n(n-1)\) (B) \(n(n-1)^{2}\) (C) \(n^{2}(n-1)\) (D) \(n^{2}(n-1)^{2}\)

If the letters of the word MOTHER are written in all possible orders and these words are written out as in a dictionary, then the rank of the word MOTHER is (A) 240 (B) 261 (C) 308 (D) 309

The number of divisors a number 38808 can have, excluding 1 and the number itself is (A) 70 (B) 72 (C) 71 (D) None of these

The number of positive integral solutions of \(15

The number of different 7 digit numbers that can be written using only the three digits 1,2 and 3 with the condition that the digit 2 occurs twice in each number is (A) \({ }^{7} P_{2} 2^{5}\) (B) \({ }^{7} C_{2} 2^{5}\) (C) \({ }^{7} C_{2} 5^{2}\) (D) None of these

Let \(S\) be the set of all functions from the set \(A\) to the set A. If \(n(A)=k\) then \(n(S)\) is (A) \(k !\) (B) \(k^{k}\) (C) \(2^{k}-1\) (d) \(2^{k}\)

The number of ways in which thirty five apples can be distributed among 3 boys so that each can have any number of apples, is (A) 1332 (B) 666 (C) 333 (D) None of these

The number of non-negative solutions of \(x_{1}+x_{2}+x_{3}\) \(+, \ldots,+x_{n} \leq n\) (where \(n\) is positive integer) is (A) \({ }^{2 n} C_{n}-1\) (B) \({ }^{2 n-1} C_{n}-1\) (C) \({ }^{2 n+1} C_{n-1}\) (D) \({ }^{2 n-1} C_{n-1}-1\)

Eleven animals of a circus have to be placed in eleven cages one in each cage. If 4 of the cages are too small for 6 of the animals, then the number of ways of caging the animals is (A) 304800 (B) 504800 (C) 604800 (D) None of these

If \(n\) is even and \({ }^{n} C_{0}<{ }^{n} C_{1}<{ }^{n} C_{2}<\ldots\left\langle{ }^{n} C_{r}>{ }^{n} C_{r+1}>\ldots>{ }^{n} C_{n}\right.\) then \(r=\) (A) \(\frac{n}{2}\) (B) \(\frac{n-1}{2}\) (C) \(\frac{n-2}{2}\) (C) \(\frac{n+2}{2}\)

In a network of railways, a small island has 15 stations. The number of different types of tickets to be printed for each class, if every station must have tickets for other station, is (A) 230 (B) 210 (C) 340 (D) None of these

The number of ordered pairs \((m, n), m, n \in\\{1,2, \ldots,\), \(50\\}\) such that \(6^{n}+9^{m}\) is a multiple of 5 is (A) 6250 (B) 1250 (C) 1875 (D) None of these

\(A\) set contains \((2 n+1)\) elements. The number of subsets of the set which contain at most \(n\) elements is (A) \(2^{n}\) (B) \(2^{n+1}\) (C) \(2^{2 n-1}\) (D) \(2^{2 n}\)

There are \(n\) concurrent lines and another line parallel to one of them. The number of different triangles that will be formed by the ( \(n+1\) ) lines, is (A) \(\frac{(n-1) n}{2}\) (B) \(\frac{(n-1)(n-2)}{2}\) (C) \(\frac{n(n+1)}{2}\) (D) \(\frac{(n+1)(n+2)}{2}\)

An \(n\)-digit number is a positive number with exactly \(n\) digits. Nine hundred distinct \(n\)-digit numbers are to be formed using only the three digits 2,5 and \(7 .\) The smallest value of \(n\) for which this is possible is (A) 6 (B) 7 (C) 8 (D) 9

If all permutations of the letters of the word \(A G A I N\) are arranged as in dictionary, the forty ninth word is (A) \(\mathrm{NAAGI}\) (B) \(\mathrm{NAGAI}\) (C) NAAIG (D) \(\mathrm{NAIAG}\)

The number of ways of choosing \(n\) objects out of \((3 n+1)\) objects of which \(n\) are identical and \((2 n+1)\) are distinct, is (A) \(2^{2 n}\) (B) \(2^{2 n+1}\) (C) \(2^{2 n}-1\) (D) None of these

In a group of boys, two boys are brothers and in this group 6 more boys are there. In how many ways they can sit if the brothers are not to sit along with each other (A) 4820 (B) 1410 (C) 2830 (D) None of these

If \(20 \%\) of three subsets (i.e., subsets containing exactly three elements) of the set \(A=\left\\{a_{1}, a_{2}, \ldots, a_{n}\right\\}\) contain \(a_{1}\), then the value of \(n\) is (A) 15 (B) 16 (C) 17 (C) 18

The number of two digit numbers which are of the form \(x y\) with \(y

The total number of ways in which a beggar can be given at least one rupee from four \(25 \mathrm{p}\). coins, three \(50 \mathrm{p}\). coins and 2 one rupee coins is (A) 54 (B) 53 (C) 51 (D) 48

\(A\) student is allowed to select atmost \(n\) books from a collection of \((2 n+1)\) books. If the total number of ways in which he can select books is 63 , then \(n=\) (A) 4 (B) 3 (C) 7 (D) 8

How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even positions? (A) 16 (B) 36 (C) 60 (D) 180

In a certain test there are \(n\) questions. In this test \(2^{k}\) students gave wrong answers to at least \((n-k)\) questions, where \(k=0,1,2, \ldots, n .\) If the total number of wrong answers is 4095 , then value of \(n\) is (A) 11 (B) 12 (C) 13 (D) 15

The number of permutations of the letters \(a, b, c, d\) such that \(b\) does not follow \(a, c\) does not follow \(b\), and \(d\) does not follow \(c\), is (A) 12 (B) 14 (C) 13 (D) 11

If \(S=\sum_{r=0}^{m}{\underline{\phantom{xx}}}^{n+r} C_{k}\), then (A) \(S+{ }^{n} C_{k+1}={ }^{n+m} C_{k+1}\) (B) \(S+{ }^{n} C_{k+1}={ }^{n+m+1} C_{k+1}\) (C) \(S+{ }^{n} C_{k}={ }^{n+m} C_{k}\) (D) None of these

The number of ways of dividing 15 men and 15 women into 15 couples, each consisting of a man and a woman, is (A) 1240 (B) 1840 (C) 1820 (D) 2005

Suman writes letters to his five friends. The number of ways can be letters be placed in the envelopes so that atleast two of them are in the wrong envelopes are (A) 119 (B) 120 (C) 125 (D) None of these

Statement 1: The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is \({ }^{9} C_{3}\). Statement 2: The number of ways of choosing any 3 places from 9 different places is \({ }^{9} \mathrm{C}_{3}\). (A) Statement 1 is false, Statement 2 is true (B) Statement \(I\) is true, Statement 2, is true; Statement 2 is the correct explanation for Statement 1 (C) Statement 1 is true; Statement 2 is true; Statement 2 is not as correct explanation for Statement 1 (D) Statement 1 is true, Statement 2 is false

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

The number of 4 -digit numbers with distinct digits is (A) 504 (B) 4536 (C) 4634 (D) 5040

In a shop there are five types of ice-creams available. A child buys six ice- creams. Statement 1: The number of different ways the child can buy the six ice-creams is \({ }^{10} C_{5}\). Statement 2: The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging \(6 \mathrm{~A}\) 's and \(4 \mathrm{~B}\) 's in a row. (A) Statement 1 is false, Statement 2 is true (B) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1 (C) Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation for Statement 1 (D) Statement 1 is true, Statement 2 is false

How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two \(S\) are adjacent? (A) \(8 \cdot{ }^{6} C_{4} \cdot{ }^{7} C_{4}\) (B) \(6 \cdot 7 \cdot{ }^{8} C_{4}\) (C) \(6 \cdot 8 \cdot{ }^{7} C_{4}\) (D) \(7 \cdot{ }^{6} \mathrm{C}_{4} \cdot{ }^{8} \mathrm{C}_{4}\)