PERMUTATION

Permutation

1) Prove

 a) n(n-1) = ⁿ⁻²Pᵣ₋₂= ⁿPᵣ

b)  ¹⁰P₃ = ⁹P₃ + 3.⁹P₂

Find the value of

1) ⁿ⁺¹P₃ = 10. ⁿ⁻¹P₂ 

2) ⁿP₄: ⁿP₆ = 1:2

3) ²ⁿ⁺¹Pₙ₋₁ : ²ⁿ⁻¹Pₙ = 3:5

4) 5. ⁴Pᵣ = 6.⁵Pᵣ₋₁ 

5) ⁵Pᵣ =  ⁶Pᵣ₋₁

6) ⁹P₅+ 5. ⁹P₄= ¹⁰Pᵣ

 7) ⁿ⁺ʳ⁺²P₂= 132 and  ⁿ⁻ʳ⁺²P₂= 20. find the value of n and r.

1) There are 8 different trains run between Kolkata and Shantiniketan. In how many ways can a man travel from Kolkata to Shantiniketan in one train and return in a a different train ?     56

2) There are 12 stations on a railway line line. How many different kinds of tickets of second class must be printed in order that a pessenger may go  from any one station to any other.                                   132

3) There are 6 way to enter in a hall. In how many ways can three men enter into that hall through different entrances?             120

4) Find, without assuming any formula for permutation, the number of permutation of 35 different things taken 13 at a time  (write theanswer in the factorial form).            35!/13!

5) In how many ways can 7 boys and three girls be arranged in a row so that no two girls will come together?                 1440

6) In how many ways the letter of the word 'CHEMISTRY' be arranged taken 4 letters at a time ?                                 3024

7) In how many ways can the letters of the word DELHI arranged taken all ?             120

8) How many numbers of four digits can be formed with the digits 3,5,7,9 when each of the the numbers consists of different digits?                              24

9) How many different permutations can be made by taking all the letters of the following words

a) CONTACT.                       1260

b) COLLEGE.                       1260

10) How many numbers of 6 digits can be formed with the digits of the number 2,3,4,5,3,2 ?

                                            180

11) There are 5 copies of one Book, 4 copies of each of two books, 6copies of each of three books and 1 copy of each of 8 books. In how many ways can all the books be arranged? ( write the answer in factorial form) 

                  39!/(5! .(4!)²(6!)²)

12) In the basis of the result of the annual examination three prizes, first, second and third, are given in every class of a school.  there are 10 students in a class. In how many can the prizes be given in that class?          720

13) In how many ways can three prizes -- 1 for recitation, one for sports and one for regular attendance be given to 10 students ?                   1000

14)  In how many ways can the result (win, defeat, draw) of  five successive matches be decided ?

                                              243

15) A dice is thrown thrice. How many different outcomes are possible ? In how many ways the result of 3 throw will be different ?                      216, 120

16) Find the number of permutations taking 5 things at a time from 10 different things, each of which can be taken upto five times repeatedly (without help of any formula)       100000

17) How many numbers of 4 digits can be formed with the digit 1, 2, 3, 4, 5, 6,7  where digits can be used more than once?

                                             2401

18) How many numbers of 5 digits can be formed with the digits 0, 2, 5, 7, no digit being repeated in any number?  96

19) How many numbers lying between between  3000 and 4000 can be formed with the digit 0,1,2,3, 4, where repetition of  digits is allowed?  How many of them are odd ?                 124,50

20)  How many numbers of five digits can be made with at least one repeated digit ?           62784

21) There are 8 questions in a question paper. In how many ways can a student answer 5 question where the answes of two particular questions

a) never occurs                      720

b) always occur                     2400

22) How many numbers of six digits can be formed using the digits of the number 577224 ? How many of these numbers so formed are even?        360,180

23) How many numbers greater than 5000 can be formed with four of the digits 3, 4, 5, 6, 7 when no digit is repeated?         72

24) If none of the figure 2, 4, 5, 7, 8, 0  be repeated, how many different numbers of 4 digits can be formed with them ?        300

25) How many even numbers of 5 digits can be formed with the digits 0, 1,2,3, 4, each digit not occurring more than once in each number? How many of them are divisible by 4 ?               60, 30

26) How many positive numbers positive numbers 3 digits and divisible by 5, can be formed when the digits of each number different from each other? 136

27)  Find the sum of all the numbers which are formed by the digits 2,3,4,5 taking all together. Each digits does not occur more than once in each number. 93324

28) How many arrangements can be made by the letters of the word ORANGE? How many of  these will begin with O? how many of these will not and with E ?                 720, 120, 600

29) How many arrangements can be made by taking all the letters of the word TRIANGLE without changing the order of vowels?

                                               6720

30) In how many ways can the letters of the word SUNDAY be arranged taken all together where the letters S, N and D are in this order ?                              120

31) Find the number of ways in which the letters of the word  DRAUGHT can be arranged so that the vowels are always together.                          1440

32) In how many ways can the letter of the world CONSTANT be arranged so that the two vowels always occur together?     2520

33) In how many ways can 37 different books be arranged on a shelf so that two particular books are never together?       35.36!

34) In how many ways can the letters of the word ALGEBRA be arranged so that two A are not together ?                     1800

35) In How many ways can the letter of the word ORION be arranged so that the two consonants do not come together?                                36

37) How many arrangements can be made out of the letters of the word COMMITTEE  taking all at a time, such that four vowels do not together?                   43200

38) How many different words can be formed taking all the letters of the word BALLOONS in which 2 L will not come together ?                          900

39) Find the rank of the word NAME when its letters are arranged as in a dictionary.  20

40) Everyday there are 6 periods of a class. In how many ways 5 different subjects can be arranged daily?              5. 6!/2!

41)  How many different arrangements can be made by the letters of the word ARRANGE so that 

I) Two R's never come together

Ii)  two A's will be together but two R's do not come together.

iii) two A's never come together and also two R's never come come together.     900, 240, 660

42) In how many ways can the letters of the word STRANGE be  arranged so that vowels may occupy only the odd places?

43) In a rail compartment there are two benches on two sides of it. Five can seat on each of the benches. In how many ways can a group of 6 boys and 4 girls can seat on each of the benches. In how many ways can a group of 6 boys and 4 girls can sit if the girls always sit at the end of the bench ?                  17280

44) How many 5 digited telephone numbers with pairwise distinct digits can be formed ?

                                           9⁵

45) In how many ways can the letter of the word BANANA be arranged?                        60

46)  how many different permutation can be made out of the letters of the expression x³y²z⁴ when written in full length?   

                                          1260

47) In how many ways can 10 paise and five coins of 5 paise can be arranged in a line so that two coins of 5 paise do not come together ?                           462 

48) The number of different messages by 5 signals with three dots and two dashes is:.   10

49) At the end of conference of each of the representatives exchange their signatures with the others. If the total number of signatures be 420, how many representatives were there where  in the conference ?                21