# Class 11 RD Sharma Solutions- Chapter 16 Permutations – Exercise 16.2 | Set 1

### Question 1. In a class, there are 27 boys and 14 girls. The teacher wants to select 1 boy and 1 girl to represent the class in a function. In how many ways can the teacher make this selection?

**Solution:**

Given:Total number of boys = 27Total number of girls = 14

So, ways to select a boy =

^{27 }P_{1 }= 27Ways to select a girl =

^{14}P_{1}= 14Ways for selecting a pair of 1 boy, 1 girl = 27 x 14 = 378

### Question 2. A person wants to buy one fountain pen, one ball pen, and one pencil from a stationery shop. If there are 10 fountain pen varieties, 12 ball pen varieties, and 5 pencil varieties, in how many ways can he select these articles?

**Solution:**

Given:Total number of fountain pen = 10Total number of ball pen = 12

Total number of fountain pencil = 5

Person wants to buy only one fountain pen, one ball pen, and one pencil

So, ways to select a pen =

^{10}P_{1 }= 10Ways to select a ball pen =

^{12}P_{1 }= 12Ways to select a pencil =

^{5}P_{1}= 5Ways for selecting the desired triplet = 10 x 12 x 5 = 600

### Question 3. From Goa to Bombay there are two routes; air, and sea. From Bombay to Delhi there are three routes; air, rail, and road. From Goa to Delhi via Bombay, how many kinds of routes are there?

**Solution:**

Given:From Goa to Bombay two routes = air, and seaFrom Bombay to Delhi there are three routes = air, rail, and road

So, the routes from Goa to Bombay =

^{2}P_{1}= 2Routes from Bombay to Delhi =

^{3}P_{1}= 3Total different routes from Goa to Delhi = 2 x 3 = 6

### Question 4. A mint prepares metallic calendars specifying months, dates, and days in the form of monthly sheets (one plate for each month). How many types of calendars should it prepare to serve for all the possibilities in future years?

**Solution:**

We need to find the different total number of calendars so that all the years can be represented by any one of these.

Case 1:Leap yearA leap year may start with any of 7 possible days (Monday to Sunday) = 7 options

Case 2:Ordinary yearAn ordinary year may start with any of 7 possible days (Monday to Sunday) = 7 options

Total calendars = 7 + 7 = 14

### Question 5. There are four parcels and five post-offices. In how many different ways can the parcels be sent by registered post?

**Solution:**

Given:Total number of parcels = 4total number of post-offices = 5

Each of the four parcels have 5 options of post-offices.

So, each parcel can be sent in

^{5}P_{1 }ways.Hence, the total ways =

^{5}P_{1}x^{5}P_{1}x^{5}P_{1}x^{5}P_{1}= 5 x 5 x 5 x 5 = 625

### Question 6. A coin is tossed five times, and outcomes are recorded. How many possible outcomes are there?

**Solution:**

Each toss can result in

^{2}P_{1 }= 2 ways.Five tosses = 2 x 2 x 2 x 2 x 2 = 32 ways of outcomes

### Question 7. In how many ways can an examine answer a set of ten true/false type questions?

**Solution:**

For answering one question:

^{2 }P_{1 }= 2 waysFor answering 10 questions: 2 x 2 x 2 x……2 (10 times) = 2

^{10 }= 1024 possibilities

### Question 8. A letter lock consists of three rings each marked with 10 different letters. In how many ways it is possible to make an unsuccessful attempt to open the lock?

**Solution:**

Number of possibilities for a single ring = 10 ways

For 3 rings: 10 x 10 x 10 = 1000 ways

Out of these, 1 way will be the correct password

So, the number of unsuccessful attempts = 1000 – 1 = 999

### Question 9. There are 6 multiple choice questions in an examination. How many sequences of answers are possible, if the first three questions have 4 choices each and the next three have 2 each?

**Solution:**

Possible sequences for first 3 questions =

^{4}P_{1 }x^{4}P_{1 }x^{4}P_{1 }=_{ }4 x 4 x 4 = 64Possible sequences for next 3 questions =

^{2}P_{1 }x^{2}P_{1 }x^{2}P_{1}= 2 x 2 x 2 = 8Total possibilities = 64 x 8 = 512

### Question 10. There are 5 books on Mathematics and 6 books on Physics in a book shop. In how many ways can a student buy:

### (i) a Mathematics book and a Physics book?

### (ii) either a Mathematics book or a Physics book?

**Solution:**

Given:Total number of Mathematics book = 5Total number of Physics book = 6

(i)Number of ways of buying Mathematics book =^{5}P_{1 }= 5Number of ways of buying Physics book =

^{6}P_{1 }= 6Total possibilities = 5 x 6 = 30

(ii)Number of ways of buying a book (can be any Maths or Physics) =^{11}P_{1}= 11

### Question 11. Given 7 flags of different colors, how many different signals can be generated if a signal requires the use of two flags, one below the other?

**Solution:**

Ways to select 2 flags out of 7 =

^{7}P_{2}= 7x(7 – 1)/2 = 21Ways of generating different signals from these 2 selected flags = 2

(say, A and B are the colors selected then A can be above and B below; and vice versa so 2 ways)

Total distinct signals possible = 21 x 2 = 42

### Question 12. A team consists of 6 boys and 4 girls, and other has 5 boys and 3 girls. How many single matches can be arranged between the two teams when a boy plays against a boy, and a girl plays against a girl?

**Solution:**

Case 1:A boy plays against a boySelect a boy from team 1 and a boy from team 2

Team 1:

^{6}P_{1}= 6Team 2:

^{5}P_{1}= 5Total ways of a boy playing against a boy = 6 x 5 = 30

Case 2:A girl plays against a girlSelect a girl from team 1 and a boy from team 2

Team 1:

^{4}P_{1}= 4Team 2:

^{3}P_{1}= 3Total ways of a girl playing against a girl = 4 x 3 = 12

Total ways of signal matches = Ways of boy playing against boy + Ways of girl playing against girl

= 30 + 12

= 42

### Question 13. Twelve students compete in a race. In how many ways first three prizes be given?

**Solution:**

Number of ways of selecting 3 winners =

^{12}P_{3}= 12 x 11 x 10 / (3 x 2 x 1) = 220For 3 winners selected, different ways of assigning the position

For first position we have 3 possibilities of people,

then for second we have 2 possibilities (other than the one already given first position)

and for third we have 1 possibility (other than the ones declared second and first positions)

So, 3 x 2 x 1 = 6 possibilities of assigning these 3 positions to the three selected people

Total ways of giving 3 prizes = No. of ways of selecting 3 people x Assigning 3 positions to the 3 people

= 220 x 6

= 1320

### Question 14. How many A.P.â€™s with 10 terms are there whose first term is in the set {1, 2, 3} and whose common difference is in the set {1, 2, 3, 4, 5}?

**Solution:**

Number of ways of selecting first term =

^{3}P_{1}= 3Number of ways of selecting common difference =

^{5}P_{1}= 5Total different A.P. series = 3 x 5 = 15

### Question 15. From among the 36 teachers in a college, one principal, one vice-principal and the teacher-in-charge are to be appointed. In how many ways can this be done?

**Solution:**

Number of ways of selecting 3 people =

^{36}P_{3}= 36 x 35 x 34 / (3 x 2 x 1)

= 7140

For 3 people selected, different ways of assigning the posts

For the principal post we have 3 possibilities of people,

then for vice-principal post we have 2 possibilities (other than the one already given principal post)

and for teacher-in-charge we have 1 possibility (other than people declared principal and vice-principal)

So, 3 x 2 x 1 = 6 possibilities of assigning these 3 posts to the three selected people

Total ways of assigning 3 posts = No. of ways of selecting 3 people x Assigning 3 posts to 3 people

= 7140 x 6

= 42840

### Question 16. How many three-digit numbers are there with no digit repeated?

**Solution:**

Ways of selecting 100th place =

^{9}P_{1}= 9 (Selecting from all digits except 0)Ways of selecting 10th place =

^{9}P_{1}= 9 (Selecting from all digits except the digit placed at 100th position)Ways of selecting unit pace =

^{8}P_{1}= 8 (Selecting from all digits except those at 100th and 10th places)Total 3-digit numbers possible with no digit repeated = 9 x 9 x 8 = 648

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