Permutations and Combination — Mathematics textbook solutions

3.2.1 Addition Principle

2 q

Worked Examples

Worked · 2

3.2.1 SolvedEx.1 $A restaurant offers five types of fruit juices and three types of milk shakes. If a customer wants to order a drink, how many choices does the customer have?$
3.2.1 SolvedEx.2 $Consider an experiment of drawing a card from a pack of 52 playing cards. What is the number of ways in which the drawn card is a spade or a club?$

3.2.2 Multiplication Principle

2 q

Worked Examples

Worked · 2

3.2.2 SolvedEx.1 $Samadhan Bhojanalay offers a thali that has four items: roti, rice, vegetable and dal. Following choices are available and one option is to be selected for each item. Roti: chapati, tandoor roti Rice: plain rice, jeera rice, dal khichadi Vegetable: dum aloo, paneer masala, mixed veg Dal: dal fry, dal tadka How many different menus are possible?$
3.2.2 SolvedEx.2 $A company decides to label each of its different products with a code that consists of two letters followed by three digits. How many different products can be labeled in this way?$

3.3 Invariance Principle

3 q

Solved Examples

Worked · 3

3.3 SolvedEx.1 $A \to D \to N \to E \to B$
3.3 SolvedEx.2 $Suppose 5 chocolates of different type are to be distributed among 4 children and there is no condition on how many chocolates a child can get (including zero.) How many different ways are possible for doing so?$
3.3 SolvedEx.3 $How many even numbers can be formed using the digits 2, 3, 7, 8 so that the number formed is less than 1000?$

Exercise 3.1

16 q

Ex 3.1 Q1 $A teacher wants to select the class monitor in a class of 30 boys and 20 girls. In how many ways can the monitor be selected if the monitor must be either a girl or a boy?$
Ex 3.1 Q2 $A Signal is generated from 2 flags by putting one flag above the other. If 4 flags of different colours are available, how many different signals can be generated?$
How many two letter words can be formed using letters from the word SPACE, when repetition of letters (i) is allowed, (ii) is not allowed?
Ex 3.1 Q3(i) $Repetition of letters is allowed.$
Ex 3.1 Q3(ii) $Repetition of letters is not allowed.$
How many three-digit numbers can be formed from the digits 0, 1, 3, 5, 6 if repetitions of digits (i) are allowed, (ii) are not allowed?
Ex 3.1 Q4(i) $Repetitions of digits are allowed.$
Ex 3.1 Q4(ii) $Repetitions of digits are not allowed.$
Ex 3.1 Q5 $How many three-digit numbers can be formed using the digits 2, 3, 4, 5, 6 if digits can be repeated?$
Ex 3.1 Q6 $A letter lock contains 3 rings and each ring containing 5 different letters. Determine the maximum number of false trials that can be made before the lock is opened?$
Ex 3.1 Q7 $In a test, 5 questions are of the form 'state, true or false'. No student has got all answers correct. Also, the answer of every student is different. Find the number of students appeared for the test.$
Ex 3.1 Q8 $How many numbers between 100 and 1000 have 4 in the units place?$
Ex 3.1 Q9 $How many numbers between 100 and 1000 have the digit 7 exactly once?$
Ex 3.1 Q10 $How many four digit numbers will not exceed 7432 if they are formed using the digits 2, 3, 4, 7 without repetition?$
Ex 3.1 Q11 $If numbers are formed using digits 2, 3, 4, 5, 6 without repetition, how many of them will exceed 400?$
Ex 3.1 Q12 $How many numbers formed with the digits 0, 1, 2, 5, 7, 8 will fall between 13 and 1000 if digits can be repeated?$
Ex 3.1 Q13 $A school has three gates and four staircases from the first floor to the second floor. How many ways does a student have to go from outside the school to his classroom on the second floor?$
Ex 3.1 Q14 $How many five-digit numbers formed using the digit 0, 1, 2, 3, 4, 5 are divisible by 5 if digits are not repeated?$

3.4 Factorial Notation

4 q

Worked Examples

Worked · 4

3.4 SolvedEx.1 $6!$
3.4 SolvedEx.2 $(7-3)! \ne 7! - 3!$
3.4 SolvedEx.3 $n$
3.4 SolvedEx.4 $\frac{12!}{5!\,7!} + \frac{12!}{6!\,6!} = \frac{13!}{6!\,7!}$

Exercise 3.2

41 q

Evaluate:
Ex 3.2 Q1(i) $8!$
Ex 3.2 Q1(ii) $10!$
Ex 3.2 Q1(iii) $10! - 6!$
Ex 3.2 Q1(iv) $(10 - 6)!$
Compute:
Ex 3.2 Q2(i) $\frac{12!}{6!}$
Ex 3.2 Q2(ii) $\left(\frac{12}{6}\right)!$
Ex 3.2 Q2(iii) $(3 \times 2)!$
Ex 3.2 Q2(iv) $3! \times 2!$
Ex 3.2 Q2(v) $\frac{9!}{3!\,6!}$
Ex 3.2 Q2(vi) $\frac{6! - 4!}{4!}$
Ex 3.2 Q2(vii) $\frac{8!}{6! - 4!}$
Ex 3.2 Q2(viii) $\frac{8!}{(6-4)!}$
Write in terms of factorials
Ex 3.2 Q3(i) $5 \times 6 \times 7 \times 8 \times 9 \times 10$
Ex 3.2 Q3(ii) $3 \times 6 \times 9 \times 12 \times 15$
Ex 3.2 Q3(iii) $6 \times 7 \times 8 \times 9$
Ex 3.2 Q3(iv) $5 \times 10 \times 15 \times 20$
Evaluate : $\frac{n!}{r!\,(n-r)!}$ for
Ex 3.2 Q4(i) $n = 8,\ r = 6$
Ex 3.2 Q4(ii) $n = 12,\ r = 12$
Ex 3.2 Q4(iii) $n = 15,\ r = 10$
Ex 3.2 Q4(iv) $n = 15,\ r = 8$
Find $n$ , if
Ex 3.2 Q5(i) $\frac{n}{8!} = \frac{3}{6!} + \frac{1}{4!}$
Ex 3.2 Q5(ii) $\frac{n}{6!} = \frac{4}{8!} + \frac{3}{6!}$
Ex 3.2 Q5(iii) $\frac{1!}{n!} = \frac{1!}{4!} - \frac{4}{5!}$
Ex 3.2 Q5(iv) $(n+1)! = 42 \times (n-1)!$
Ex 3.2 Q5(v) $(n+3)! = 110 \times (n+1)!$
Find $n$ , if:
Ex 3.2 Q6(i) $\frac{(17-n)!}{(14-n)!} = 5!$
Ex 3.2 Q6(ii) $\frac{(15-n)!}{(13-n)!} = 12$
Ex 3.2 Q6(iii) $\frac{n!}{3!\,(n-3)!} : \frac{n!}{5!\,(n-5)!} = 5 : 3$
Ex 3.2 Q6(iv) $\frac{n!}{3!\,(n-3)!} : \frac{n!}{5!\,(n-7)!} = 1 : 6$
Ex 3.2 Q6(v) $\frac{(2n)!}{7!\,(2n-7)!} : \frac{n!}{4!\,(n-4)!} = 24 : 1$
Ex 3.2 Q7 $\frac{n!}{r!\,(n-r)!} + \frac{n!}{(r-1)!\,(n-r+1)!} = \frac{(n+1)!}{r!\,(n-r+1)!}$
Ex 3.2 Q8 $\frac{9!}{3!\,6!} + \frac{9!}{4!\,5!} = \frac{10!}{4!\,6!}$
Ex 3.2 Q9 $\frac{(2n)!}{n!} = 2^{n}(2n-1)(2n-3)\cdots 5 \cdot 3 \cdot 1$
Simplify
Ex 3.2 Q10(i) $\frac{(2n+2)!}{(2n)!}$
Ex 3.2 Q10(ii) $\frac{(n+3)!}{(n^{2}-4)(n+1)!}$
Ex 3.2 Q10(iii) $\frac{1}{n!} - \frac{1}{(n-1)!} - \frac{1}{(n-2)!}$
Ex 3.2 Q10(iv) $n\,[\,n! + (n-1)!\,] + n^{2}(n-1)! + (n+1)!$
Ex 3.2 Q10(v) $\frac{n+2}{n!} - \frac{3n+1}{(n+1)!}$
Ex 3.2 Q10(vi) $\frac{1}{(n-1)!} + \frac{1-n}{(n+1)!}$
Ex 3.2 Q10(vii) $\frac{1}{n!} - \frac{3}{(n+1)!} - \frac{n^{2}-4}{(n+2)!}$
Ex 3.2 Q10(viii) $\frac{n^{2}-9}{(n+3)!} + \frac{6}{(n+2)!} - \frac{1}{(n+1)!}$

3.5.1 Permutations of Distinct Objects

6 q

Solved Examples

Worked · 6

3.5.1 SolvedEx.1 ${}^{5}P_{2}$
3.5.1 SolvedEx.2 $How many different ways are there to arrange letters of the word 'WORLD'? How many of these arrangements begin with the letter R? How many arrangements can be made taking three letters at a time?$
3.5.1 SolvedEx.3 $How many three digit numbers can be formed from the digits 2, 4, 5, 6, 7 if no digit is repeated?$
3.5.1 SolvedEx.4 $How many numbers can be formed with the digits 3, 4, 6, 7, 8 taken all at a time? Find the sum of all such numbers.$
3.5.1 SolvedEx.5 $A teacher has 2 different books on English, 3 different books on Physics, and 4 different books on Mathematics. These books are to be placed in a shelf so that all books on any one subjects are together. How many different ways are there to do this?$
3.5.1 SolvedEx.6 ${}^{n}P_{5} = 42\times {}^{n}P_{3}$

3.5.2 Permutations with Repetition

5 q

Solved Examples

Worked · 5

It is required to arrange 8 books on a shelf. Find the number of ways to do this if two specified books are
3.5.2 SolvedEx.1(i) $always together$
3.5.2 SolvedEx.1(ii) $never together.$
3.5.2 SolvedEx.2 $In how many ways can 7 examination papers be arranged so that papers 6 and 7 are never together?$
A family of 3 brothers and 5 sisters is to be arranged for a photograph in such a ways that,
3.5.2 SolvedEx.3(i) $all brothers sit together.$
3.5.2 SolvedEx.3(ii) $no two brothers sit together.$

Exercise 3.3

34 q

Ex 3.3 Q1 ${}^{n}P_{6} : {}^{n}P_{3} = 120:1$
Ex 3.3 Q2 ${}^{(m+n)}P_{2} = 56$
Ex 3.3 Q3 ${}^{12}P_{r-2} : {}^{11}P_{r-1} = 3:14$
Ex 3.3 Q4 $(n+1)\left({}^{n}P_{r}\right) = (n-r+1)\left[{}^{(n+1)}P_{r}\right]$
How many 4 letter words can be formed using letters in the word MADHURI if
Ex 3.3 Q5(a) $letters can be repeated$
Ex 3.3 Q5(b) $letters cannot be repeated.$
Determine the number of arrangements of letters of the word ALGORITHM if
Ex 3.3 Q6(a) $vowels are always together$
Ex 3.3 Q6(b) $no two vowels are together.$
Ex 3.3 Q6(c) $consonants are at even positions.$
Ex 3.3 Q6(d) $O is the first and T is the last letter.$
Ex 3.3 Q7 $In a group photograph, 6 teachers are in the first row and 18 students are in the second row. There are 12 boys and 6 girls among the students. If the middle position is reserved for the principal and if no two girls are together, find the number of arrangements.$
Find the number of ways so that letters of the word HISTORY can be arranged as,
Ex 3.3 Q8(a) $Y and T are together$
Ex 3.3 Q8(b) $Y is next to T.$
Ex 3.3 Q8(c) $there is no restriction$
Ex 3.3 Q8(d) $begin and end with vowel$
Ex 3.3 Q8(e) $end in ST$
Ex 3.3 Q8(f) $begin with S and end with T$
Ex 3.3 Q9 $Find the number of arrangements of the letters in the word SOLAPUR so that consonants and vowels are placed alternately$
Find the number of 4-digit numbers that can be formed using the digits 1, 2, 4, 5, 6, 8 if
Ex 3.3 Q10(a) $digits can be repeated$
Ex 3.3 Q10(b) $digits cannot be repeated$
Ex 3.3 Q11 $How many numbers can be formed using the digits 0, 1, 2, 3, 4, 5 without repetition so that resulting numbers are between 100 and 1000?$
Find the number of 6-digit numbers using the digits 3, 4, 5, 6, 7, 8 without repetition. How many of these numbers are
Ex 3.3 Q12(a) $divisible by 5,$
Ex 3.3 Q12(b) $not divisible by 5.$
Ex 3.3 Q13 $A code word is formed by two different English letters followed by two non-zero distinct digits. Find the number of such code words. Also, find the number of such code words that end with an even digit.$
Ex 3.3 Q14 $Find the number of ways in which 5 letters can be posted in 3 post boxes if any number of letters can be posted in a post box.$
Find the number of arranging 11 distinct objects taken 4 at a time so that a specified object
Ex 3.3 Q15(a) $always occurs$
Ex 3.3 Q15(b) $never occurs.$
In how many ways can 5 different books be arranged on a shelf if
Ex 3.3 Q16(i) $there are no restrictions$
Ex 3.3 Q16(ii) $2 books are always together$
Ex 3.3 Q16(iii) $2 books are never together$
3 boys and 3 girls are to sit in a row. How many ways can this be done if
Ex 3.3 Q17(i) $there are no restrictions$
Ex 3.3 Q17(ii) $there is a girl at each end$
Ex 3.3 Q17(iii) $boys and girls are at alternate places$
Ex 3.3 Q17(iv) $all boys sit together$

3.5.3 Permutations with Identical Objects

2 q

Solved Examples

Worked · 2

3.5.3 SolvedEx.1 $Find he number of permutations of the letters of the word UBUNTU.$
3.5.3 SolvedEx.2 $How many arrangements can be made, with the letters of the word CALCULATOR? In how many of these arrangements, vowels occur together?$

Exercise 3.4

21 q

Find the number of permutations of letters in each of the following words.
Ex 3.4 Q1(i) $DIVYA$
Ex 3.4 Q1(ii) $SHANTARAM$
Ex 3.4 Q1(iii) $REPRESENT$
Ex 3.4 Q1(iv) $COMBINE$
Ex 3.4 Q1(v) $BALBHARATI$
Ex 3.4 Q2 $You have 2 identical books on English, 3 identical books on Hindi, and 4 identical books on Mathematics. Find the number of distinct ways of arranging them on a shelf .$
A coin is tossed 8 times. In how many ways can we obtain.
Ex 3.4 Q3(a) $4 heads and 4 tails?$
Ex 3.4 Q3(b) $at least 6 heads?$
Ex 3.4 Q4 $A bag has 5 red, 4 blue, and 4 green marbles. If all are drawn one by one and their colours are recorded, how many different arrangements can be found?$
Ex 3.4 Q5 $Find the number of ways of arranging letters of the word MATHEMATICAL How many of these arrangements have all vowels together?$
Ex 3.4 Q6 $Find the number of different arrangements of letters in the word MAHARASHTRA. How many of these arrangements have (a) letters R and H never together? (b) all vowels together?$
Ex 3.4 Q7 $How many different words are formed if the letters R is used thrice and letters S and T are used twice each?$
Ex 3.4 Q8 $Find the number of arrangements of letters in the word MUMBAI so that the letter B is always next to A.$
Ex 3.4 Q9 $Find the number of arrangements of letters in the word CONSTITUTION that begin and end with N.$
Ex 3.4 Q10 $Find the number of different ways of arranging letters in the word ARRANGE. How many of these arrangement do not have the two R's and two A's together?$
Ex 3.4 Q11 $How many distinct 5 digit numbers can be formed using the digits 3, 2, 3, 2, 4, 5.$
Ex 3.4 Q12 $Find the number of distinct numbers formed using the digits 3, 4, 5, 6, 7, 8, 9, so that odd positions are occupied by odd digits.$
Ex 3.4 Q13 $How many different 6-digit numbers can be formed using digits in the number 659942? How many of them are divisible by 4?$
Ex 3.4 Q14 $Find the number of distinct words formed from letters in the word INDIAN. How many of them have the two N's togethers?$
Find the number of different ways of arranging letters in the word PLATOON if.
Ex 3.4 Q15(a) $the two O's are never together.$
Ex 3.4 Q15(b) $consonants and vowels occupy alternate positions.$

3.5.4 Circular Permutations

10 q

Solved Examples

Worked · 10

In how many ways can 8 students be arranged at a round table so that 2 particular students are together, if
3.5.4 SolvedEx.1(i) $students are arranged with respect to each other? (That is the seats are not numbered.)$
3.5.4 SolvedEx.1(ii) $students are arranged with respect to the table? (That is the seats are numbered serially)$
3.5.4 SolvedEx.2 $In how many ways 6 men and 3 women can be seated at a round table so that every man has woman by his side.$
3.5.4 SolvedEx.3 $Find the number of ways in which 12 different flowers can be arranged in a garland so that 4 particular flowers are always together$
3.5.4 SolvedEx.4 $How many necklaces of 12 bead each, can be made from 18 beads of different colours?$
3.5.4 SolvedEx.5 $Three boys and three girls are to be seated around a table in a circle. Among them, the boy X does not want any girl as neighbour and girl Y does not want any boy as neighbour. How many such distinct arrangements are possible?$
In how many different arrangements can 6 gentlemen and 6 ladies sit around a table if
3.5.4 SolvedEx.6(i) $there is no restriction.$
3.5.4 SolvedEx.6(ii) $no two ladies sit side by side?$
In how many different ways can 4 married couples occupy seats around a circular table if
3.5.4 SolvedEx.7(i) $Spouses sit opposite to each other?$
3.5.4 SolvedEx.7(ii) $Men and women alternate?$

Exercise 3.5

12 q

Ex 3.5 Q1 $In how many different ways can 8 friends sit around a table?$
Ex 3.5 Q2 $A party has 20 participants. Find the number of distinct ways for the host to sit with them around a circular table. How many of these ways have two specified persons on either side of the host?$
Delegates from 24 countries participate in a round table discussion. Find the number of seating arrangements where two specified delegates are.
Ex 3.5 Q3(a) $always together,$
Ex 3.5 Q3(b) $never together.$
Ex 3.5 Q4 $Find the number of ways for 15 people to sit around the table so that no two arrangements have the same neighbours.$
Ex 3.5 Q5 $A committee of 10 members sits around a table. Find the number of arrangements that have the president and the vice president together.$
Five men, two women, and a child sit around a table. Find the number of arrangements where the child is seated
Ex 3.5 Q6(a) $between the two women.$
Ex 3.5 Q6(b) $between two men.$
Ex 3.5 Q7 $Eight men and six women sit around a table. How many of sitting arrangements will have no two women together?$
Ex 3.5 Q8 $Find the number of seating arrangements for 3 men and 3 women to sit around a table so that exactly two women are together$
Ex 3.5 Q9 $Four objects in a set of ten objects are alike. Find the number of ways of arranging them in a circular order.$
Ex 3.5 Q10 $Fifteen persons sit around a table. Find the number of arrangements that have two specified persons not sitting side by side.$

3.6.1 Properties of Combinations

15 q

Solved Examples

Worked · 15

Find the value of
3.6.1 SolvedEx.1(i) ${}^{7}C_{3}$
3.6.1 SolvedEx.1(ii) ${}^{10}C_{7}$
3.6.1 SolvedEx.1(iii) ${}^{52}C_{3}$
3.6.1 SolvedEx.2 $n$
There are $n$ points in a plane. Find the number of straight lines and triangles that can be obtained by joining points on a plane if
3.6.1 SolvedEx.3(i) $no three points are collinear$
3.6.1 SolvedEx.3(ii) $p$
Four cards are drawn from a pack of 52 playing cards. In how many different ways can this be done? How many selections will contain
3.6.1 SolvedEx.4(i) $exactly one card of each suit?$
3.6.1 SolvedEx.4(ii) $all cards of the same suit?$
3.6.1 SolvedEx.4(iii) $all club cards?$
3.6.1 SolvedEx.4(iv) $at least one club card?$
3.6.1 SolvedEx.4(v) $three kings and one queen?$
3.6.1 SolvedEx.4(vi) $three black and one red cards?$
3.6.1 SolvedEx.5 $n$
3.6.1 SolvedEx.6 $r$
3.6.1 SolvedEx.7 ${}^{8}C_{r}$

Exercise 3.6

44 q

Find the value of
Ex 3.6 Q1(a) ${}^{15}C_{4}$
Ex 3.6 Q1(b) ${}^{80}C_{2}$
Ex 3.6 Q1(c) ${}^{15}C_{4} + {}^{15}C_{5}$
Ex 3.6 Q1(d) ${}^{20}C_{16} - {}^{19}C_{16}$
Find $n$ if
Ex 3.6 Q2(a) ${}^{6}P_{2} = n\, {}^{6}C_{2}$
Ex 3.6 Q2(b) ${}^{2n}C_{3} : {}^{n}C_{2} = 52 : 3$
Ex 3.6 Q2(c) ${}^{n}C_{n-3} = 84$
Ex 3.6 Q3 $r$
Find $n$ and $r$ if.
Ex 3.6 Q4(a) ${}^{n}P_{r} = 720$
Ex 3.6 Q4(b) ${}^{n}C_{r-1} : {}^{n}C_{r} : {}^{n}C_{r+1} = 20 : 35 : 42$
Ex 3.6 Q5 ${}^{n}P_{r} = 1814400$
Ex 3.6 Q6 ${}^{n}C_{r-1} = 6435$
Ex 3.6 Q7 $Find the number of ways of drawing 9 balls from a bag that has 6 red balls, 8 green balls, and 7 blue balls so that 3 balls of every colour are drawn.$
Ex 3.6 Q8 $Find the number of ways of selecting a team of 3 boys and 2 girls from 6 boys and 4 girls.$
Ex 3.6 Q9 $After a meeting, every participant shakes hands with every other participants. If the number of handshakes is 66, find the number of participants in the meeting.$
Ex 3.6 Q10 $If 20 points are marked on a circle, how many chords can be drawn?$
Find the number of diagonals of an $n$ -sided polygon. In particular, find the number of diagonals when.
Ex 3.6 Q11(a) $n = 10$
Ex 3.6 Q11(b) $n = 15$
Ex 3.6 Q11(c) $n = 12$
Ex 3.6 Q11(d) $n = 8$
Ex 3.6 Q12 $There are 20 straight lines in a plane so that no two lines are parallel and no three lines are concurrent. Determine the number of points of intersection.$
Ten points are plotted on a plane. Find the number of straight lines obtained by joining these points if
Ex 3.6 Q13(a) $no three points are collinear.$
Ex 3.6 Q13(b) $four points are collinear.$
Find the number of triangles formed by joining 12 points if
Ex 3.6 Q14(a) $no three points are collinear$
Ex 3.6 Q14(b) $four points are collineas.$
Ex 3.6 Q15 $A word has 8 consonants and 3 vowels. How many distinct words can be formed if 4 consonants and 2 vowels are chosen?$
Find $n$ if,
Ex 3.6 Q16(i) ${}^{n}C_{8} = {}^{n}C_{12}$
Ex 3.6 Q16(ii) ${}^{23}C_{3n} = {}^{23}C_{2n+3}$
Ex 3.6 Q16(iii) ${}^{21}C_{6n} = {}^{21}C_{(n^{2}+5)}$
Ex 3.6 Q16(iv) ${}^{2n}C_{r-1} = {}^{2n}C_{r+1}$
Ex 3.6 Q16(v) ${}^{n}C_{n-2} = 15$
Ex 3.6 Q17 $x$
Ex 3.6 Q18 $r$
Ex 3.6 Q19 $\sum_{r=1}^{4} {}^{(21-r)}C_{4}$
Find the differences between the greatest values in the following:
Ex 3.6 Q20(a) ${}^{14}C_{r}$
Ex 3.6 Q20(b) ${}^{13}C_{r}$
Ex 3.6 Q20(c) ${}^{15}C_{r}$
Ex 3.6 Q21 $In how many ways can a boy invite his 5 friends to a party so that at least three join the party?$
Ex 3.6 Q22 $A group consists of 9 men and 6 women. A team of 6 is to be selected. How many of possible selections will have at least 3 women?$
Ex 3.6 Q23 $A committee of 10 persons is to be formed from a group of 10 women and 8 men. How many possible committees will have at least 5 women? How many possible committees will have men in majority?$
Ex 3.6 Q24 $A question paper has two sections. section I has 5 questions and section II has 6 questions. A student must answer at least two question from each section among 6 questions he answers. How many different choices does the student have in choosing questions?$
Ex 3.6 Q25 $There are 3 wicketkeepers and 5 bowlers among 22 cricket players. A team of 11 players is to be selected so that there is exactly one wicketkeeper and at least 4 bowlers in the team. How many different teams can be formed?$
Five students are selected from 11. How many ways can these students be selected if.
Ex 3.6 Q26(a) $two specified students are selected?$
Ex 3.6 Q26(b) $two specified students are not selected?$

Miscellaneous Exercise 3

31 q

(I) Select the correct answer

Practice · 10

Misc I Q1 $A college offers 5 courses in the morning and 3 in the evening. The number of ways a student can select exactly one course, either in the morning or in the evening.$
1. A. $5$
2. B. $3$
3. C. $8$
4. D. $15$
Misc I Q2 $A college has 7 courses in the morning and 3 in the evening. The possible number of choices with the student if he wants to study one course in the morning and one in the evening is-.$
1. A. $21$
2. B. $4$
3. C. $42$
4. D. $10$
Misc I Q3 $In how many ways can 8 Indians and, 4 American and 4 Englishmen can be seated in a row so that all person of the same nationality sit together?$
1. A. $3!\,8!$
2. B. $3!\,4!\,8!\,4!$
3. C. $4!\,4!$
4. D. $8!\,4!\,4!$
Misc I Q4 $In how many ways can 10 examination papers be arranged so that the best and the worst papers never come together?$
1. A. $9 \times 8!$
2. B. $8 \times 8!$
3. C. $9 \times 9!$
4. D. $8 \times 9!$
Misc I Q5 $In how many ways 4 boys and 3 girls can be seated in a row so that they are alternate.$
1. A. $12$
2. B. $288$
3. C. $144$
4. D. $256$
Misc I Q6 $Find the number of triangles which can be formed by joining the angular points of a polygon of 8 sides as vertices.$
1. A. $16$
2. B. $56$
3. C. $24$
4. D. $8$
Misc I Q7 $A question paper has two parts, A and B, each containing 10 questions. If a student has to choose 8 from part A and 5 from part B, In how many ways can he choose the questions?$
1. A. $320$
2. B. $750$
3. C. $40$
4. D. $11340$
Misc I Q8 $There are 10 persons among whom two are brothers. The total number of ways in which these persons can be seated around a round table so that exactly one person sits between the brothers, is equal to:$
1. A. $2! \times 7!$
2. B. $2! \times 8!$
3. C. $3! \times 7!$
4. D. $3! \times 8!$
Misc I Q9 $The number of arrangements of the letters of the word BANANA in which two N's do not appear adjacently$
1. A. $80$
2. B. $60$
3. C. $40$
4. D. $100$
Misc I Q10 $The number of ways in which 5 male and 2 female members of a committee can be seated around a round table so that the two females are not seated together is$
1. A. $840$
2. B. $600$
3. C. $720$
4. D. $480$

(II) Answer the following

Practice · 21

Misc II Q1 $r$
Misc II Q2 $How many words can be formed by writing letters in the word CROWN in different order?$
Misc II Q3 $40^{\text{th}}$
Misc II Q4 $Capital English alphabet has 11 symmetric letters that appear same when looked at in a mirror. These letters are A, H, I, M, O, T, U, V, W, X, and Y. How many symmentric three letter passwords can be formed using these letters?$
Misc II Q5 $How many numbers formed using the digits 3,2,0,4,3,2,3 exceed one million?$
Misc II Q6 $Ten students are to be selected for a project from a class of 30 stdudents. There are 4 students who want to be together either in the project or not in the project. Find the number of possible selections.$
Misc II Q7 $A student finds 7 books of his interest, but can borrow only three books. He wants to borrow Chemistry part II book only if Chemistry Part I can also be borrowed. Find the number of ways he can choose three books that he wants to borrow.$
Misc II Q8 $30 objects are to be divided in three groups containing 7,10,13 objects. Find the number of distinct ways for doing so.$
Misc II Q9 $A student passes an examination if he secures a minimum in each of the 7 subjects. Find the number of ways a student can fail.$
Misc II Q10 $Nine friends decide to go for a picnic in two groups. One group decides to go by car and the other group decides to go by train. Find the number of different ways of doing so if there must be at least 3 friends in each group.$
Misc II Q11 $A hall has 12 lamps and every lamp can be switched on independently. Find the number of ways of illuminating the hall.$
Misc II Q12 $How many quadratic equations can be formed using numbers from 0,2,4,5 as coefficients if a coefficient can be repeated in an equation.$
Misc II Q13 $-$
Misc II Q14 $A question paper has 6 questions. How many ways does a student have to answer if he wants to solve at least one question?$
Misc II Q15 $Find the number of ways of dividing 20 objects in three groups of sizes 8,7,and 5.$
Misc II Q16 $There are 4 doctors and 8 lawyers in a panel. Find the number of ways for selecting a team of 6 if at least one doctor must be in the team.$
Misc II Q17 $Four parallel lines intersect another set of five parallel lines. Find the number of distinct parallelograms formed.$
There are 12 distinct points A,B,C,.....,L, in order, on a circle. Lines are drawn passing through each pair of points
Misc II Q18(i) $How many lines are there in total.$
Misc II Q18(ii) $How many lines pass through D.$
Misc II Q18(iii) $How many triangles are determined by lines.$
Misc II Q18(iv) $How many triangles have on vertex C.$