Chapter 7

From 6 ditferent novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on the shelf so that the dictionary is always in the middle. Then 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

One ticket is selected at random from 50 tickets numbered \(00,01,02, \ldots, 49\). Then the probability that the sum of the digits on the selected ticket is 8 , given that the product of these digits is zero, equals (A) \(\frac{1}{14}\) (B) \(\frac{1}{7}\) (C) \(\frac{5}{14}\) (D) \(\frac{1}{50}\)

In a certain test, \(a_{i}\) students gave wrong answers to at least \(i\) questions where \(i=1,2,3, \ldots, k\). No student gave more than \(k\) wrong answers. The total number of wrong answers given is (A) \(a_{1}+a_{2}+\ldots+a_{k}\) (B) \(a_{1}+a_{2}+\ldots+a_{k-1}\) (C) \(a_{1}+a_{2}+\ldots+a_{k+1}\) (D) None of these

A gentleman invites 13 guests to a dinner and places 8 of them at one table and remaining 5 at the other, the tables being round. The number of ways he can arrange the guests is (A) \(\frac{11 !}{40}\) (B) \(9 !\) (C) \(\frac{12 !}{40}\) (D) \(\frac{13 !}{40}\)

There are stalls for 10 animals in a ship. The number of ways the shipload can be made if there are cows, calves and horses to be transported, animals of each kind being not less than 10 , is (A) 59049 (B) 49049 (C) 69049 (D) None of these

In an examination a candidate has to pass in each of the papers to be successful. If the total number of ways to fail is 63 , how many papers are there in the examination? (A) 6 (B) 8 (C) 14 (D) None of these

If \(A\) denotes the property that two elements of \(A=\\{1\), \(5,9,13 \ldots, 1093\\}\) add up to 1094 , then the maximum number of elements in \(A\) can be (A) 126 (B) 136 (C) 137 (D) 138

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 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

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 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) \({ }^{7} P_{2} 2^{2}\) (B) \({ }^{7} C_{2} 2^{5}\) (C) \({ }^{7} C_{2} 5^{5}\) (D) None of these

There are 10 points in a plane of which no three points are collinear and 4 points are concyclic. The number of different circles that can be drawn through at least 3 of these points is (A) 116 (B) 120 (C) 117 (D) None of these

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

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) NAAGI (B) \(N A G A I\) (C) NAAIG (D) \(N A I A G\)

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

For \(x \in R\), let \([x]\) denotes the greatest integer \(\leq x\), then the value of \(\left[-\frac{1}{3}\right]+\left[-\frac{1}{3}-\frac{1}{100}\right]+\left[-\frac{1}{3}-\frac{2}{100}\right]\) \(+\ldots+\left[-\frac{1}{3}-\frac{99}{100}\right]\) is (A) \(-100\) (B) \(-123\) (C) \(-135\) (D) \(-153\)

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

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 \(S=\sum_{r=0}^{m}{\underline{\phantom{xx}}}^{m+r} C_{k}\), 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

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

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

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

Given 5 different green dyes, 4 different blue dyes and 3 different red dyes, the number of combinations of dyes that can be chosen by taking at least one green and one blue dye is (A) 248 (B) 120 (C) 3720 (D) 465

Number of points having position vector \(a \hat{i}+b \hat{j}+c \hat{k}\) where \(a, b, c \in\\{1,2,3,4,5\\}\) such that \(2^{a}+3^{b}+5^{c}\) is divisible by 4 is (A) 140 (B) 70 (C) 100 (D) None of these

\({ }^{2 n} \mathrm{C}_{r}(0 \leq r \leq 2 n)\) is greatest when \(r\) is equal to (A) \(\frac{n}{2}\) (B) \(\frac{n+1}{2}\) (C) \(r=n\) (D) None of these

The number of even numbers greater than 100 that can be formed by the digits \(0,1,2,3\) (no digit being repeated) is (A) 20 (B) 30 (C) 40 (D) None of these

The number of positive numbers less than 1000 and divisible by 5 (no digit being repeated) is (A) 150 (B) 154 (C) 166 (D) None of these

In a certain city, all telephone numbers have six digits, the first two digits always being 41 or 42 or 46 or 62 or 64 . The number of telephone numbers having all six digits distinct is (A) 8400 (B) 7200 (C) 9200 (D) None of these

The sum of five digit numbers which can be formed with the digits \(3,4,5,6,7\) using each digit only once in each arrangement, is (A) 5666600 (B) 6666600 (C) 7666600 (D) None of these

The sum of all the numbers that can be formed by writing all the digits \(3,2,3,4\) only once is (A) 39996 (B) 49996 (C) 57776 (D) None of these

The sum of all numbers greater than 10000 formed by using the digits \(1,3,5,7,9\), no digit being repeated in any number, is (A) 4666600 (B) 5666600 (C) 6666600 (D) None of these

The sum of all numbers greater than 1000 formed by using the digits \(0,1,2,3\), no digit being repeated in any number, is (A) 38664 (B) 48664 (C) 58664 (D) None of these

The number of four digit numbers that can be formed from the digits \(0,1,2,3,4,5\) with at least one digit repeated is (A) 420 (B) 560 (C) 780 (D) None of these

The number of odd numbers lying between 40000 and 70000 that can be made from the digits \(0,1,2,4,5,7\) if digits can be repeated in the same number is (A) 864 (B) 932 (C) 766 (D) None of these

A table has provision for 7 seats, 4 being on one side facing the window and 3 being on the opposite side. The number of ways in which 7 people can be seated at the table if 2 people, \(\mathrm{X}\) and \(\mathrm{Y}\), must sit on the same side, is (A) 3260 (B) 2160 (C) 3350 (D) None of these

There are four oranges, five apples and six mangoes in a fruit basket. The number of ways in which a person can make a selection of fruits among the fruits in the basket, is (A) 210 (B) 330 (C) 209 (D) None of these

The number of zeros at the end of \(100 !\) is (A) 36 (B) 18 (C) 24 (D) None of these

The largest integer \(n\) such that \(33 !\) is divisible by \(2^{n}\) is (A) 30 (B) 31 (C) 32 (D) None of these

The number of non-negative integral solutions of \(x_{1}+\) \(x_{2}+x_{3}+4 x_{4}=20\) is (A) 436 (B) 536 (C) 602 (D) None of these

The product of \(r\) consecutive positive integers is divisible by (A) \(r !\) (B) \((r-1) !\) (C) \((r+1) !\) (D) None of these

The number of ordered triplets of positive integers which are solutions of the equation \(x+y+z=100\) is (A) 5081 (B) 6005 (C) 4851 (D) None of these

The number of words that can be formed, with the letters of the work 'Pataliputra' without changing the relative order of the vowels and consonants, is (A) 3600 (B) 4200 (C) 3680 (D) None of these

On a new year day every student of a class sends a card to every other student. The postman delivers 600 cards. The number of students in the class are (A) 42 (B) 34 (C) 25 (D) None of these

For any positive integers \(m, n\) (with \(n \geq m\) ), let and \(\left(\begin{array}{l}n \\ m\end{array}\right)={ }^{n} C_{m}\), then \(\left(\begin{array}{l}n \\ m\end{array}\right)+\left(\begin{array}{l}n-1 \\\ m\end{array}\right)+\left(\begin{array}{l}n-2 \\\ m\end{array}\right)+\ldots+\left(\begin{array}{c}m \\ m\end{array}\right)=\) (A) \(\left(\begin{array}{l}n+1 \\ m\end{array}\right)\) (B) \(\left(\begin{array}{l}n+1 \\ m+1\end{array}\right)\) (C) \(\left(\begin{array}{c}n \\ m+1\end{array}\right)\) (D) None of these

The number of ways of choosing \(m\) coupons out of an unlimited number of coupons bearing the letters \(A, B\) and \(C\) so that they cannot be used to spell the word \(B A C\), is (A) \(3\left(2^{m}-1\right)\) (B) \(3\left(2^{m-1}-1\right)\) (C) \(3\left(2^{m}+1\right)\) (D) None of these

The number of ways in which 16 identical things can be distributed among 4 persons if each person gets at least 3 things, is (A) 33 (B) 35 (C) 38 (D) None of these