PERMUTATIONS & COMBINATIONS

Q1. There are 16 buses running between Meerut and Delhi. In how many ways could the round trip
from Meerut can be made, if the return was made on (i) the same bus, (ii) a different bus. (16,240)

Q2. How many 5 letter code words are possible using 8 letters of the English alphabet, if no letter
can be repeated. (6720)

Q3. 7 pictures are to be arranged from left to right on a wall of an art gallery for display. How many
arrangements are possible. (5040)

Q4. How many 7 digit telephone no. can be created using the digits 0 to 9, if each number starts with
71 and no digit appears more than once. (6720)

Q5. How many 4 digit nos. can be formed from the digits 1,2,3,4,5,and 6 assuming (i) repetition of
digits allowed (ii) repetition not allowed. (64, 360)

Q6. A coin is tossed two times and the outcomes are noted. How many possible outcomes are
there? How many possible outcomes if the coin is tossed 6times? 8 times? 21 times? (22, 26, 28, 221)

Q7. Given 6 flags of different colours, how many different signals can be generated if the signal
requires the use of three flag one below the other? (120)

Q8. How many numbers can be formed from the digits 1,2,3,4,5 if repetition of digits is not
allowed?(120)

Q9. Evaluate (i) 6! – 4! (ii) 4!+4! (696, 30)

(iii) Is 3! + 4! = 7! (iv) Calculate 3!.4!. Is 3!.4! = 12!? (144) (v) Compute 6!/2! (360)
16! 6! 6!
Q10. (i) 2!(16-2)! (ii) 3(3!) (iii) 4!3! (120,40, 5)

Q11. (i) If nP4 = 360, find the value of n. (6) (ii) if P(11,r) = P(12,r-1), find r. (9)
(iii) if P(n-1,3):P(n,4) = 1:9, find n. (9) (iv) if P(2n-1, n): P(2n+1, n-1) = 22:7, find n (10)
11(n-1) n+3
Q12. (i) if P(15,r-1):P(16,r-2)=3:4, find r. (14) (ii) n+5Pn+1= 2
. Pn. find n. (6,7)

Q13. Find the number of 4 digit numbers that can be formed using the digits 1,2,3,4,5 if no digit is
used more than once in a number. How many of these numbers will be odd? (120,72)

Q14. How many odd numbers of three digits each can be made with the digit 1,2,3,4,5,6,7 if no digit
is repeated? (60)

Q15 How many different four digit number can be formed from the digits 2,3,4 and 6 if each digit is
used only once in a number? How many of these numbers (i) end in 4 (ii) end in 3 (iii) end in 3 or 6
(24,6,6,12)

Q16. (i) Prove that (i)nPn = 2.nPn-2 (ii) 10P3=9P3 + 3.9P2

Q17. Prove that nPr = n-1Pr + r.n-1Pr-1

Q18. How many 3 letter words can be made using the letter of the word ORIENTAL? (336)

Q19. Find number of different 8 letter arrangements that can be made from the letters of the word
DAUGHTER so that all the vowels do not come together? (36000)

Q20. How many words can be formed out of the word ARTICLE so that vowels occupy even places?
(144)
 PERMUTATIONS & COMBINATIONS

Q21. How many words can be formed by the letters of the word ALLAHABAD that starts with A?
(3360)

Q22. From a pool of 12 candidates, in how many ways can we select president, vice president,
secretary and treasurer, if each of the 12 candidates can hold any office? (11880)

Q23. In how many ways can 5 persons A,B,C,D and E sit around a circular table if (i) B and D next to
each other, (ii) B and D do not sit next to each other? (12,12)

Q24. Evaluate : (i) 19C17 +19C18 (ii) 25C22 – 24C21 (190,276)

Q25. Determine n and r if n-1Cr : nCr : n+1Cr = 6:9:13 [n=12,r=4]

Q26. Find the value of nC3 if 2nC3:nC2 = 44:3 [6]

Q27. There are 15 points in a plane, no three of which are collinear. Find the number of Triangle
formed by joining them. [455]

Q28. A bookshelf contains 7 different Mathematics books and five different Physics books. How
many groups of 3 mathematics and 3 Physics books can be selected? [350]

Q29. In an examination, a candidate has to select 4 from each part 8 I and II. There are 6 and 7
questions in part I and part II respectively. What is the number of possible combinations in which we
can choose the questions? [525]

Q30. In how many ways boys can choose a programme of 6 courses if 10 courses are available and 3
courses are compulsory for every student? [35]

Q31. There are 16 players. How many teams of 11 players can be selected if two particular players
are (i)always selected (ii)always rejected? [2002,364]

Q32. In how many ways 4 cards can be drawn from a pack of 52 playing cards so that there is at least
one face card? [179335]

Q33. A committee of 3 persons is to be constituted from a group of 3 men and 3 women. In how
many ways committee may be formed so as to include at least one women?[19]

Q34. A sports team of 11 students is to be constituted, choosing at least 5 from class 11TH and at
least 5 from class 12th. If there are 30 students in each class, in how many ways can the team be
constituted? [2C(30,5)C(30,6)]

Q35. In how many different ways, the letters of the word ALGEBRA can be arranged in a row if (i) the
2 A's are together, (ii) the two A's are not together [720,1800]

Q36. 5 girls and 5 boys are to be seated on a bench with the boys and girls alternating. Find the
number of ways of there sitting. In how many different ways would they sit around a circular table
so that boys and girls alternate.[28800,2880]

Q37. How many numbers greater than a million can be formed with the digits 2,3,0,3,4,2,3? [360]

Q38. In an examination a candidate is required to answer 6 out of 10 questions which are divided
into two groups each containing 5 questions and he is not permitted to attempt more than 4
questions from a group. In how many ways he can make up his choice? [200]

Q39. A committee of 6 is to be formed from 6 men and 4 women. In how many ways can this be
done if the committee contains (i) two women, (ii) at least two women? [90,185]
 PERMUTATIONS & COMBINATIONS

Q40. There are 12 points in a plane of which 5 are collinear. How many lines can be constructed by
joining two points.[57]

Q41. In question 40, how many Triangles are possible?[20]

Q42. Using digits from 0 to 9, find the numbers of (i) one digit, (ii) four digits,(iii) five digits can be
formed when repetition of digits any number of time is allowed.[9,9000,90000]

Q43. A coin is tossed 6 times at random. Find the number of ways of obtaining 4 heads and 2
tails?[15]

Q44. In how many ways can the letters of word ARRANGE be arranged to so that (i) the two R's are
never together,(ii) the two A's are together but not 2R's, (iii) neither 2R's nor 2A's are
together.[900,240,660]

Q45. In how many ways can 6 boys and 4 girls be seated in a row so that no two girls are
together.[604800]

Q46. Find the total no. of arrangements of the letters in the expression a3b2c4 when written at full
length. [1260]

Q47. How many words can be formed with the letters of the word PARALLEL so that all L’s do not
come together? [3000]
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