LAST TIME REVISION PERMUTATION

1) Find the values of 
i) 27!/24! .                 17550

ii) 12!/(6! 4!).            27720

iii) (11! - 10!)/9!.           100

iv) LCM and GCM of 6!, 7!, 8!.           8!, 6!

2) Prove, 1/9! + 1/10! + 1/11! = 122/11!

3) Convert the following products into factorials:
i) 5 x6x7x8x9.             9!/4!

ii) 3x6x9x12x 15x 18.          3⁶ x 6!

3) Find the value of n, if 
i) 1/8! + 1/9! = n/10!.               100

ii) (n + 2)! = 60(n -1)! .        3

5)  (2n)!/{3!(2n -3)!} and n!/{2!(n -2)!} are in the ratio 44 : 3, find n.    6

6) Show that (2n)!= 2ⁿ . n! {1.3.5.  (2n -1)}

7) prove that, 2.6.10.14....to n factors = (n+1)(n+2)(n+3)..... to n factors.

8) There are 17 stations from the Howrah to Bangalore on the railway line. How many kinds of different single 3rd class tickets must be printed so as to enable a passenger to travel from one station to another ?       272

9) How many words can be made using all the letters in the word MONDAY ? How many of them begin with M and do not end with Y ?    120, 96

10) How many words can be made from the letters of the word COSTING so that the vowels
i) are always together.        1440
ii) are never together.        3600
iii) may occupy only odd positions.     1440

11) Out of the letters A, B, C, p,q,r. how many different words can be formed if
i) the words always begin with a capital letter.       360
ii) the words always begin with a capital letter and end also with a capital letter ?  144

12) In how many ways can 12 examination papers be arranged so that the best and worst papers may never come together ?         10 (11)!

13) How many numbers each lying between 1000 to 10000 can be formed with the digits 1, 2, 3, 4, 5, 6, 7 (no digit being repeated in any number)?       840

14) horw many odd numbers of 6 significant figures can be formed with the digits 0,1, 2, 3, 4 and 5, none of the digits being repeated in any of the numbers so formed?  288

15) How many even numbers greater than 300 can be formed with the digits 1,2,3,4 and 5, no repetitions being allowed ?   111

16) Find the sum of the numbers which can be formed using all the five digits of 12345 only once.       3999960

17) Find the number of permutation of 20 things taken 5 at a time so that one particular thing 
i) will always occur.     465120
ii) will never occur.       1395360

18) In how many ways can the letters of the word ASSASSINATION be arranged ? In how many of these arrangements the four S's do not come together ? In how many arrangements the vowel always come together.      10810800, 10659600, 50400

19) How many numbers greater than a million can be formed with the digits 3,4,4,0,3,5,4?    360

20) How many numbers of not more than 4 digits can be formed with the digits 1, 2, 3 and 4 repetitions being allowed ?    340

21) In how many ways can 10 HS students and 7 B.Com students be arranged in a line so that no two 2 B. Com students may sit together ? 10! X ¹¹P₇

22) Find the number of permutation of n different things taken r at a time so that m(<r) particular things
i) will always occur.      ʳPₘ x ⁿ⁻ᵐPᵣ₋ₘ 
 
ii) will never occur.   ⁿ⁻ᵐᵣ

23) The first name of person consist of 8 letters in which one letter occurs more than once while the other letters are different. If the number of permutations of the letters of his name taken all at a time be 6720, find the number of times the like letter occurs.    3

24) In how many ways can 6 boys form a ring ?       120

25) In how many ways can 6 men be seated at a around table?    720

26) In how many ways can 6 beads of different colours be arranged to form a necklace ?   60

27) In how many different ways 4 letters can be put in 4 different envelopes ?   24

28) There are 6 colleges in South Calcutta. In how many ways can a man send 4 of his sons to a college so that no two of them may come in the same college ?  360

29) Find the values of n and r, when
i) ⁿ⁺ʳP₂ = 110, ⁿ⁻ʳP₂20.     8,3

ii) ⁿ⁺ʳ⁺¹P₂ =72, ⁿ⁻ʳP₂ =12.    6,2

30) Prove 33! is divisible by 2¹⁵.

31) How many different permutation can be made by taking all the letters of the word BENGALI ?     5040

32) There are 25 stations on a railway line. How many different kinds of single 3rd class tickets must be printed in order that it may be possible to travel from one station to another ?     600

33) How many different permutation can be made by taking 5 of the letters of the word THURSDAY?    6720

34) In how many ways can the letters of the word LOGARITHM be arranged? How many of these arrangements begin with L ? How many begin with L and do not end with M?       362880, 40320, 35280

35) Find how many words can be formed of the word FAILURE,  the four vowels always coming together .    676

36) six papers are set in an examination, of which 2 are mathematical. In how many different orders can the paper be arranged so that 
i) the two mathematical papers are together.     240
ii) the two mathematical papers are not consecutive ?     480

37) In how many ways 3 boys and 5 girls can be arranged in a row so that all the three boys are together ?      4320

38) Show that the number of ways in which n books may be arranged on a shelf so that to particular books shall not be together is (n -2)(n-1)!

39) Find the number of numbers each less than 999 and divisible by 2 which can be formed with the digits 2, 3 ,4, 5 , 6 and 7, no digit occurring more than once in any number.       78

40) Find the number of numbers less than 1000 and divisible by 5 which can be formed with the digit 0, 1, 2, 3, 4, 5 , 6, 7, 8, 9 ; each digits occurring not more than once in each number.       154

41) In how many ways can 5 commerce and 5 science students be arranged in a row so that the commerce and the science students are placed alternative ?    2880

42) In how many of the permutation at 12 things taken 3 at a time will one particular thing 
i) always occur.        330
ii) never occur ?     990

43) In how many ways of the permutation 12 things taken 6 at a time will 3 particular things 
i) always occur .           604800
ii) never occur ?          60480

44) Find the number of different permutations that can be made out of the letters of the following words :
i) COMMERCE.      5040
ii) ACCOUNTANT.      226800
iii) ENGINEERING.            11!/(3!3!2!2!)
iv) STATISTICS.             50400
v) SUCCESS.                420


45) In how many ways can the letters of the name GAVASKAR be arranged so that 3 a's may come together ?       720

46) Show that the letters of the word INSURANCE can be arranged in twice as many ways as the letters of to word ECONOMICS.      

47) How many different arrangements can be made out of the letters of the expression x³y²z⁴ when written at full length ?     1260

48) The first name of a person consists of 9 letters in which one letter occurs more than once and the other letters are different. If the number of permutation of the letters of his name taken all at a time be 15120, find the number of times the like letter occurs.      4

49) In how many ways can the letters of the word CONTACT be arranged
i) without changing the order of the vowels.            630
ii) without changing the relative positions of the vowels and consonants.      60
iii) without changing the positions of the vowels.            30


50) Find the sum of the numbers which can be formed by using all the four digits of 3456 only once.    119988

51) A library has 5 copies of one book, 4 copies of each of two books , 6 copies of each of three books and 1 copy of eight books. In how many ways can all the books be arranged?   39!(5!4!²6!³)

52) How many numbers greater than a lakh be formed with the digits 0,2,5,2,4,5 ?   150

53) In how many ways four prizes -- one for recitation, one for sports, one for smartness and for general proficiency be given away to 8 boys ?    8⁴

54) How many three figures number can be formed from the digits 1,2,3,4,5,6, 7, 8, 9 ? ( there is no restriction on the repetition of the digits ).        729

55) How many numbers of not more than 4 digits can be formed with 6, 7, 8 ?   120

56) In how many ways can 10 children sit in a merry-go-round relatively to form a another ?      9!

57) In how many different ways can 15 different flowers be arranged to form a garland ?  14!/2

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