Chapter 5

The number of ways in which we can get a score of 11 by throwing three dice is a. 18 b. 27 c. 45 d. 56

In how many different ways can the first 12 natural numbers be divided into three different groups such that numbers in each group are in A.P.? a. 1 b. 5 c. 6 d. 4

Fifteen identical balls have to be put in five different boxes. Each box can contain any number of balls. The total number of ways of putting the balls into the boxes so that each box contains at least two balls is equal to a. \({ }^{9} C_{5}\) b. \({ }^{10} \mathrm{C}_{5}\) c. \({ }^{6} C_{5}\) d. \({ }^{10} C_{b}\)

The number of ways to give 16 different things to three persons \(A, B, C\) so that \(B\) gets one more than \(A\) and \(C\) gets two more than \(B\), is a. \(\frac{16 !}{4 ! 5 ! 7 !}\) b. \(4 ! 5 ! 7 !\) c. \(\frac{16 !}{3 ! 5 ! 8 !}\) d. none of these

The number of ways in which we can distribute \(m n\) students equally among \(m\) sections is given by a. \(\frac{(m n) !}{n !}\) b. \(\frac{(m n) !}{(n !)^{m}}\) c. \(\frac{(m n) !}{m ! n !}\) d. \((m n)^{m}\)

\(2 m\) white counters and \(2 n\) red counters are arranged in a straight line with \((m+n)\) counters on each side of a central mark. The number of ways of arranging the counters, so that the arrangements are symmetrical with respect to the central mark, is a. \({ }^{m+n} C_{m}\) b. \(\quad \stackrel{2 m+2 "}{\underline{\phantom{xx}}}^{2 m+2} C_{2 m}\) c.,\(\frac{1}{2} \frac{(m+n) !}{m ! n !}\) d. none of these

A person buys eight packets of TIDE detergent. Each packet contains one coupon, which bears one of the letters of the word TIDE. If he shows all the letters of the word TIDE, he gets one free packet. If he gets exactly one free packet, then the number of different possible combinations of the coupons is a. \({ }^{7} C_{3}\) b. \(\quad{ }^{8} C_{4}\) c. \({ }^{8} C_{3}\) d. \(4^{4}\)

There are three copies each of four different books. The number of ways in which they can be arranged in a shelf is a. \(\frac{12 !}{(3 !)^{4}}\) b. \(\frac{12 !}{(4 !)^{3}}\) c. \(\frac{21 !}{(3 !)^{+} 4 !}\) d. \(\frac{12 !}{(4 !)^{3} 3 !}\)

The number of ways in which 12 books can be put in three shelves with four on each shelf is a. \(\frac{12 !}{(4 !)^{3}}\) b. \(\frac{12 !}{(3 !)(4 !)^{3}}\) c. \(\frac{12 !}{(3 !)^{3} 4 !}\) d. none of these

The total number of ways in which \(2 n\) persons can be divided i nto \(n\) couples is a. \(\frac{2 n !}{n ! n !}\) b. \(\frac{2 n !}{(2 !)^{n}}\) c. \(\frac{2 n !}{n !(2 !)^{n}}\) d. none of these

Let \(x_{1}, x_{2}, \ldots, x_{k}\) be the divisors of positive integer ' \(n\) ' (including 1 and \(n\) ). If \(x_{1}+x_{2}+\cdots+x_{k}=75\), then \(\sum_{i=1} 1 / x_{i}\) is equal to a. \(\frac{75}{n^{2}}\) b. \(\frac{75}{n}\) c. \(\frac{75}{k}\) d. none of these

Let \(A=\left(x_{1}, x_{2}, x_{3}, \ldots ; x_{7}\right\\} . B=\left\\{y_{1}, y_{2}, y_{3}\right\\} .\) The total number of functions \(f: A \rightarrow B\) that are on to and there are exactly three element \(x\) in \(A\) such that \(f(x)=y_{2}\) is equal to a. 490 b. 510 c. 630 d. none of these

The total number of ways in which \(n^{2}\) number of identical balls can be put in \(n\) numbered boxes \((1,2,3, \ldots, n)\) such that \(i^{\text {it }}\) box contains at least \(i\) number of balls is a. \({ }^{n^{2}} C_{s-1}\) b. \(\quad{ }^{n^{2}-1} C_{n-1}\) c. \(\quad \frac{\mu^{2}+u-2}{2} C_{H-1}\) d. none of these

The total number of ways in which 15 identical blankets can be distributed among four persons so that each of them gets at least two blankets is equal to a. \({ }^{10} C_{1}\) b. \(\quad{ }^{9} C_{3}\) C. \({ }^{\prime \prime} C_{3}\) d. none of these

Number of ways in which 25 identical things be distributed among five persons if each gets odd number of things is a. \({ }^{25} C_{4}\) b. \({ }^{12} C_{s}\) c. \({ }^{14} C_{10}\) d. \({ }^{13} C_{3}\)

Number of ways in which Rs. 18 can be distributed amongst four persons such that nobody receives less than Rs. 3 is a. \(4^{2}\) b. \(2^{+}\) c. \(4 !\) d. none of these

In how many ways can 17 persons depart from railway station in 2 cars and 3 autos, given that 2 particular persons depart by same car ( 4 persons can sit in a car and 3 persons can sit in an auto)? a. \(\frac{15 !}{2 ! 4 !(3 !)^{3}}\) b. \(\frac{16 !}{(2 !)^{2} 4 !(3 !)^{3}}\) c. \(\frac{17 !}{2 ! 4 !(3 !)^{3}}\) d. \(\frac{15 !}{4 !(3 !)^{3}}\)

The total number of ways of selecting six coins out of 20 onerupee coins, 10 fifty-paise coins and 7 twenty-five paise coins is a. 28 b. 56 c \(\quad{ }^{37} \mathrm{C}_{6}\) d. none of these

The total number of ways of selecting two number from the set \(\\{1,2,3,4, \ldots, 3 n\\}\) so that their sum is divisible by 3 is equal to a. \(\frac{2 n^{2}-n}{2}\) b. \(\frac{3 n^{2}-n}{2}\) c. \(2 n^{2}-n\) d. \(3 n^{2}-n\)

Among the \(8 !\) permutations of the digits \(1,2,3, \ldots, 8\), consider those arrangements which have the following property. If we take any five consecutive positions, the product of the digits in these positions is divisible by \(5 .\) The number of such arrangements is equal to a. \(7 !\) b. \(2 .(7 !)\) c. \({ }^{7} C_{4}\) d. none of these

The total number of divisors of 480 , that are of the form \(4 n+2, n \geq\) 0 , is equal to a. 2 b. 3 c. 4 d. none of these

The total number of times, the digit ' 3 ' will be written, when the integers having less than 4 digits are listed is equal to a. 300 b. 310 c. 302 d. 306

Straight lines are drawn by joining \(m\) points on a straight line to \(n\) points on another line. Then excluding the given points, the number of point of intersections of the lines drawn is (no two lines drawn are parallel and no three lines are concurrent) a. \(\frac{1}{4} m n(m-\mathrm{l})(n-1)\) b. \(\frac{1}{2} m n(m-1)(n-1)\) c. \(\frac{1}{2} m^{2} n^{2}\) d. \(\frac{1}{4} m^{2} n^{2}\)

In a polygon, no three diagonals are concurrent. If the total number of points of intersection of diagonals interior to the polygon is 70 , then the number of diagonals of the polygon is a. 20 b. 28 c. 8 . d. none of these

Two packs of 52 cards are shuffled together. The number of ways in which a man can be dealt 26 cards so that he does not get two cards of the same suit and same denomination is a. \({ }^{\$ 7} C_{26} \cdot 2^{26}\) b. \({ }^{104} C_{26}\) c. \(2 \cdot{ }^{32} C_{26}\) d. none of these

There are \((n+1)\) white and \((n+1)\) black balls each set numbered 1 to \(n+1\). The number of ways in which the balls can be arranged in a row so that the adjacent balls are of different colours is a. \((2 n+2) !\) b. \((2 n+2) ! \times 2\) c. \((n+1) ! \times 2\) d. \(2\\{(n+1) !\\}^{2}\)

The number of three-digit numbers of the form \(x y z\) such that \(x

\(A\) is a set containing ' \(n\) ' different elements. A subset \(P\) of \(A\) is chosen. The set \(A\) is reconstructed by replacing the elements of \(P\). A subset \(Q\) of \(A\) is again chosen. The number of ways of choosing \(P\) and \(Q\) so that \(P \cap Q\) contains exactly two elements is a. \({ }^{n} C_{3} \times 2^{\prime \prime}\) b. \(\quad{ }^{n} C_{2} \times 3^{n-2}\) c. \(3^{n-2}\) d. none of these

Messages are conveyed by arranging four white, one blue and three red flags on a pole. Flags of the same colour are alike. If a message is transmitted by the order in which the colours are arranged, the total number of messages that can be transmitted if exactly six flags are used is a. 45 b. 65 c. 125 d. 185

A seven-digit number without repetition and divisible by 9 is to be formed by using seven digits out of \(1,2,3,4,5,6,7,8.9 .\) The number of ways in which this can be done is a. \(9 !\) b. \(2(7 !)\) c. \(4(7 !)\) d. none of these

The number of distinct natural numbers up to a maximum of four digits and divisible by 5, which can be formed with the digits 0 , \(1,2,3,4,5,6,7,8,9\) each digit not occurring more than once in each number is a. 1246 b. 952 c. 1106 d. none of these

A man has three friends. The number of ways he can invite one friend everyday for dinner on six successive nights so that no friend is invited more than three times is a. 640 b. 320 c. 420 d. 510

There are four letters and four directed envelopes. The number of ways in which all the letters can be put in the wrong envelope is a. 8 b. 9 c. 16 d. none of these

A bag contains four one-rupee coins, two twenty-five paisa coins and five ten- paisa coins. In how many ways can an amount, not less than Re 1 be taken out from the bag? (Consider coins of the same denominations to be identical.) a. 71 b. 72 c. 73 d. 80

In a certain test, there are \(n\) questions. In the test \(2^{n-i}\) students gave wrong answers to at least \(i\) questions, where \(i=1,2, \ldots, n .\) If the total number of wrong answers given is 2047 , then \(n\) is equal to a. 10 b. 11 c. 12 d. 13

A train timetable must be compiled for various days of the week so that two trains twice a day depart for three days, one train daily for two days and three trains once a day for two days. How many different timetables can be compiled? a. 140 b. 210 c. 133 d. 72

The total number of positive integral solution of \(15