Chapter 7

The number of integers between 1 and 1000000 that have the sum of the digits 18, is (A) 25927 (B) 25827 (C) 24927 (D) None of these

The number of non-negative integral solutions to the system of equations \(x+y+z+u+t=20\) and \(x+y+\) \(z=5\) is (A) 336 (B) 346 (C) 246 (D) None of these

The number of positive integral solutions of the inequality \(3 x+y+z \leq 30\), is (A) 1115 (B) 1215 (C) 1315 (D) None of these

In a city no person has identical set of teeth and there is no person without a tooth. Also, no person has more than 32 teeth. If we disregard the shape and size of tooth and consider only the positioning of the teeth, then the maximum population of the city is (A) \(2^{32}\) (B) \(2^{32}-1\) (C) \(2^{32}+1\) (D) None of these

Eleven scientists are working on a secret project. They wish to lock up the documents in a cabinet such that cabinet can be opened if six or more scientists are present. Then, the smallest number of locks needed is (A) 460 (B) 461 (C) 462 (D) None of these

The number of numbers greater than \(10^{6}\) that can be formed using the digits of the number 2334203, if all the digits of the given number must be used, is (A) 360 (B) 420 (C) 260 (D) None of these

If ' \(n\) 'is an integer between 0 and 21 , then the minimum value of \(n !(21-n) !\) is (A) \(9 ! 2 !\) (B) \(10 ! 11 !\) (C) \(20 !\) (D) \(21 !\)

In how many ways can 20 oranges be given to four children if each child should get at least one orange? (A) 869 (B) 969 (C) 973 (D) None of these

The total number of 5 -digit numbers of different dig. its in which the digit in the middle is the largest is (A) \(\sum_{n=4}^{9}\left({ }^{n} P_{4}-{ }^{n-1} P_{3}\right)\) (B) \(\sum_{n=4}^{9}{\underline{\phantom{xx}}}^{n} P_{4}\) (C) \(\sum_{n=4}^{9}{\underline{\phantom{xx}}}^{n-1} P_{3}\) (D) None of these

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) 5 (B) 6 (C) 7 (D) 8

If \(a, b, c\) are three natural numbers in A.P. such that \(a+b+c=21\), then the possible number of values of \(a, b, c\) is (A) 13 (B) 14 (C) 15 (D) 16

There are 10 points in a plane, no three of which are in the same straight line excepting 4 , which are collinear. Then, number of (A) straight lines formed by joining them is 40 (B) triangles formed by joining them is 116 (C) straight lines formed by joining them is 45 (D) triangles formed by joining them is 120

If \(N\) is the number of positive integral solutions of \(x_{1} x_{2} x_{3} x_{4}=770 .\) Then, (A) \(N\) is divisible by 4 distinct primes (B) \(N\) is a perfect square (C) \(N\) is a perfect 4 th power (D) \(N\) is a perfect 8 th power

Let \(E=\left[\frac{1}{3}+\frac{1}{50}\right]+\left[\frac{1}{3}+\frac{2}{50}\right]+\ldots+\) up to 50 terms, then (A) \(E\) is divisible by exactly 2 primes (B) \(E\) is prime (C) \(E \geq 30\) (D) \(E \leq 35\)

The number of ways in which three numbers in A.P. can be selected from \(1,2,3, \ldots, n\) is (A) \(\frac{n(n-2)}{4}\), when \(n\) is even (B) \(\frac{1}{4}(n-1)^{2}\), when \(n\) is odd (C) \(\frac{n(n-2)}{2}\), when \(n\) is even (D) None of these

If \(n

If \(n\) objects are arranged in a row, the number of ways of selecting three of these objects so that no two adjacent objects are selected, is (A) \({ }^{n-2} C_{3}\) (B) \({ }^{n-2} C_{n-5}\) (C) \({ }^{n-3} C_{2}\) (D) \({ }^{n-3} C_{n-5}\)

If \({ }^{n} C_{r-1}=\left(k^{2}-8\right)\left({ }^{n+1} C_{r}\right)\), then \(k\) belongs to (A) \([-3,-2 \sqrt{2}]\) (B) \([-3,-2 \sqrt{2})\) (C) \([2 \sqrt{2}, 3]\) (D) \((2 \sqrt{2}, 3]\)

The number of non-negative integral solutions of \(x_{1}+x_{2}+x_{3}+x_{4} \leq n\) (where \(n\) is a positive integer) is (A) \({ }^{n+4} C_{n}\) (B) \({ }^{n+4} C_{4}\) (C) \({ }^{n+3} C_{3}\) (D) \({ }^{n+3} C_{n}\)

There are five different boxes and seven different balls. All the seven balls are to be distributed in the five boxes placed in a row so that any box can receive any number of balls. In how many ways can these balls be distributed so that no box is empty? (A) 71 (B) 16800 (C) 1775 (D) None of these

There are five different boxes and seven different balls. All the seven balls are to be distributed in the five boxes placed in a row so that any box can receive any number of balls. Suppose, all the balls are identical, then in how many ways can all these balls be distributed into these boxes? (A) 110 (B) 220 (C) 330 (D) 1440

There are five different boxes and seven different balls. All the seven balls are to be distributed in the five boxes placed in a row so that any box can receive any number of balls. In how many ways can these balls be distributed so that box 2 and box 4 contain only 1 and 2 balls, respectively? (A) 5522 (B) 8505 (C) 2305 (D) None of these

There are five different boxes and seven different balls. All the seven balls are to be distributed in the five boxes placed in a row so that any box can receive any number of balls. In how many ways can these balls be distributed such that no box is empty and ball 2 and ball 4 cannot be put in the same box? (A) 1200 (B) 15000 (C) 3800 (D) None of these

There are three pots and four coins. All these coins are to be distributed into these pots where any pot can contain any number of coins. In how many ways all these coins can be distributed if all coins are identical but all pots are different? (A) 15 (B) 16 (C) 17 (D) 81

There are three pots and four coins. All these coins are to be distributed into these pots where any pot can contain any number of coins. In how many ways all these coins can be distributed if all coins are different but all pots are identical? (A) 14 (B) 21 (C) 27 (D) None of these

There are three pots and four coins. All these coins are to be distributed into these pots where any pot can contain any number of coins. In how many ways all these coins can be distributed such that no pot is empty if all coins are identical but all pots are different? (A) 6 (B) 3 (C) 9 (D) 27

There are three pots and four coins. All these coins are to be distributed into these pots where any pot can contain any number of coins. In how many ways all these coins can be distributed if all coins are identical and two pots are also identical? (A) 1 (B) 10 (C) 9 (D) 11

There are three pots and four coins. All these coins are to be distributed into these pots where any pot can contain any number of coins. In how many ways all these coins can be distributed if out of 4 coins 2 coins are identical and all pots are different? (A) 45 (B) 27 (C) 54 (D) None of these

In the World Cup, the tournament is arranged as per the following rules: In the beginning, 16 teams are taken and divided into 2 groups of 8 teams each. Teams of each group play a match against each other in the same group. From each group, 4 top teams qualify for the next round. In the next round, two teams play each other in each group and the losing team goes out of the tournament. Then, four winning teams play for semifinal round and finally there is one final. The rules of the tournament are such that every match can result only in a win or a loss and not in a tie. The total number of matches played in the tournament is (A) 51 (B) 64 (C) 63 (D) 52

In the World Cup, the tournament is arranged as per the following rules: In the beginning, 16 teams are taken and divided into 2 groups of 8 teams each. Teams of each group play a match against each other in the same group. From each group, 4 top teams qualify for the next round. In the next round, two teams play each other in each group and the losing team goes out of the tournament. Then, four winning teams play for semifinal round and finally there is one final. The rules of the tournament are such that every match can result only in a win or a loss and not in a tie. The maximum number of matches that a team going out of the tournament in the first round can win is (A) 1 (B) 2 (C) 3 (D) 4

In the World Cup, the tournament is arranged as per the following rules: In the beginning, 16 teams are taken and divided into 2 groups of 8 teams each. Teams of each group play a match against each other in the same group. From each group, 4 top teams qualify for the next round. In the next round, two teams play each other in each group and the losing team goes out of the tournament. Then, four winning teams play for semifinal round and finally there is one final. The rules of the tournament are such that every match can result only in a win or a loss and not in a tie. The minimum number of matches that a team must win in order to qualify for the second round is (A) 4 (B) 5 (C) 6 (D) 7

$$ \begin{array}{ll} \hline \text { Column-I } & \text { Column-II } \\ \hline \text { (I) The sum of the digits in the } & \text { (A) } 286 \\ \text { unit's place of all the numbers } \\ \text { formed with the help of } 3,4,5 \\ 6 \text { taken all at a time is... } \\ \text { (II) The number of ways in which a } \\ \text { committee of } 5 \text { can be chosen } \\ \text { from } 10 \text { candidates so as to } \\ \text { exclude the youngest if it } \\ \text { includes the oldest, is... } \\ \text { (III) The number of divisors of } 9600 & \text { (C) } 196 \\ \text { including } 1 \text { and } 9600 \text { are... } \\ \text { (IV) The number of ways of } & \text { (D) } 48 \\ \text { choosing } 10 \text { balls from infinite } & \\ \text { white, red, blue and green balls } & \\ \text { is... } \\ \hline \end{array} $$

$$ \begin{array}{ll} \text { Column-I } & \text { Column-II } \\ \hline \text { (I) }{\underline{\phantom{xx}}}^{n-2} C_{r}+2 .^{n-2} C_{r-1}+ & \text { (A) } n+3 \\ { }^{n-2} C_{r-2}=\ldots \\ \text { (II) }{\underline{\phantom{xx}}}^{m} C_{r+1}+\sum_{k=m}^{n}{\underline{\phantom{xx}}}^{k} C_{r}=\ldots & \text { (B) }{\underline{\phantom{xx}}}^{n+1} C_{r+1} \\ \text { (III) If }{\underline{\phantom{xx}}}^{n} C_{n-r}+3 \cdot{ }^{n} C_{n-r+1}+3 \\ { }^{n} C_{n-r+2}+{ }^{n} C_{n-r+3}={ }^{x} \mathrm{C}_{r}, \\ \text { then } x=\ldots \\ \begin{array}{ll} \text { (IV) The total number of } \\ \text { permutations of } n \text { different } \\ \text { things taken not more than } \end{array} \\ r \text { at a time, when each thing } \\ \text { may be repeated any number } \\ \text { of times is... } \end{array} $$

Instructions: In the following questions an Assertion \((A)\) is given followed by a Reason \((R) .\) Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: In an examination consisting of 9 papers, a candidate has to pass in more papers than the numbers of papers in which he fails in order to be successful. The number of ways in which he can be unsuccessful 256 . Reason: \({ }^{n} C_{0}+{ }^{n} C_{1}+{ }^{n} C_{2}+\ldots+{ }^{n} C_{n}=2^{n}\)

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 choices available to him is (A) 140 (B) 196 (C) 280 (D) 346

If \(^{n} C_{r}\) denotes the number of combinations of \(n\) things taken \(r\) at a time, then the expression \({ }^{n} C_{r+1}+{ }^{n} C_{r-1}+\) \(2 \times{ }^{n} C_{r}\) equals (A) \({ }^{n+2} C_{r}\) (B) \({ }^{n+2} C_{r+1}\) (C) \({ }^{n+1} C_{r}\) (D) \({ }^{n+1} C_{r+1}\)

How many ways are there to arrange the letters in the word GARDEN with the vowels in alphabetical order? (A) 120 (B) 480 (C) 360 (D) 240

The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is (A) 5 (B) \({ }^{8} C_{3}\) (C) \(3^{8}\) (D) 21

If the letters of 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 (A) 601 (B) 600 (C) 603 (D) 602

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 elected. If a voter votes for at least one candidate, then the number of ways in which he can vote is (A) 5040 (B) 6210 (C) 385 (D) 1110

How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two \(\mathrm{S}\) 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) \(7 \cdot{ }^{6} C_{4} \cdot{ }^{8} C_{4}\)

From 6 different 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

There are two urns. Urn \(I\) has 3 distinct red balls and Urn \(I I\) has 9 distinct blue balls. From each urn two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is (A) 36 (B) 66 (C) 108 (D) 3

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 \({ }^{9} C_{3}\). (A) Statement- 1 is true, Statement- 2 is true; Statement- 2 is not a correct explanation for Statement- 1 (B) Statement- 1 is true, Statement- 2 is false. (C) Statement- 1 is false, Statement- 2 is true. (D) Statement- 1 is true, Statement- 2 is true; Statement-2 is a correct explanation for Statement- 1

Assuming the balls to be identical except for difference in colors, 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

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 the value of \(n\) is (A) 5 (B) 10 (C) 8 (D) 7

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) 192 (B) 120 (C) 72 (D) 216

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) \(58^{\text {th }}\) (B) \(46^{\text {th }}\) (C) \(59^{\text {th }}\) (D) \(52^{\text {nd }}\)