54. Permutations and Combinations Problems Part 1

  1. If \(^nP_4 = 360,\) find \(n\).
  2. If \(^nP_3 = 9240,\) find \(n\).
  3. If \(^{10}P_r = 720,\) find \(r\).
  4. If \(^{2n + 1}P_{n - 1}:^{2n - 1}P_n = 3:5,\) find \(n\).
  5. Prove that \(^1P_1 + 2.^2P_2 + 3.^3P_3 + \ldots + n.^nP_n = ^{n + 1}P_{n + 1} - 1\)
  6. How many numbers of four digits can be formed with digits \(1, 2, 3, 4\) and \(5\) if repetition of digits is not allowed?
  7. How many numbers between \(400\) and \(1000\) can be made with the digits \(2, 3, 4, 5, 6\) and \(0\)?
  8. Find the number of numbers between \(300\) and \(3000\) that can be formed with the digits \(0, 1, 2, 3, 4\) and \(5\) with no repetitions.
  9. How many even numbers of four digits can be formed with the digits \(0, 1, 2, 3, 4, 5\) and \(6\) with no repetitions.
  10. How many numbers of four digits greater than \(2300\) can be formed with digits \(0, 1, 2, 3, 4, 5\) and \(6\) with no repetitions.
  11. How many numbers can be formed by using any number of digits \(0, 1, 2, 3\) and \(4\) with no repetitions.
  12. How many numbers of four digits can be formed with the digits \(1, 2, 3\) and \(4\). Find the sum of those numbers.
  13. Find the sum of all four digit numbers that can be formed with the digits \(0, 1, 2\) and \(3\).
  14. Find the sum of all the \(4\) digit numbers that can be formed with the digits \(1, 2, 2\) and \(3\).
  15. A person has to send invitations to \(6\) friends. In how many ways can he send invitations to them if he has three servants.
  16. In how many ways \(3\) prizes can be given away to \(7\) boys with each is eligible for any number of the prizes.
  17. A telegraph has \(5\) arms and each arm is capable of \(4\) distinct positions, including the position of rest. What is the total number of signals that can be made.
  18. A letter lock consists of three rings each marked with \(10\) different letters. In how many ways is it possible to make an unsuccessful attempts to open the lock?
  19. How many numbers greater than \(1000\) but less than \(4000\) can be formed with the digits \(0, 1, 2, 3\) and \(4\) with repetitions allowed.
  20. In how many ways can \(8\) Indians, \(4\) Americans and \(4\) Englishmen be seated in a row so that all persons of the same nationality sit together?
  21. There are \(20\) books of which \(4\) are single volume and the other are books of \(8, 5\) and \(3\) volumes. In how many ways can all these books be arranged on a shelf so that volumes of the same book are not separated?
  22. A library has two books each having three copies and three other books each having two copies each. In how many ways can all these books be arranged in a shelf so that copies of same book are not separated?
  23. In how many ways can \(10\) examination papers be arranged so that the best and worst papers never come together?
  24. There are \(5\) boys and \(3\) girls. In how many ways can they be seated in a row so that not all girls sit together?
  25. In how many ways can \(7\) I.A. and \(5\) I. Sc. students can be seated in a row so that no two of the I. Sc. students may sit together?
  26. In a class there are \(7\) boys and \(3\) girls. In how many different ways can then be arranged in a row such that no two of the three girls are consecutive?
  27. In how many ways \(4\) boys and \(4\) girls can be seated in a row so that boys and girls are alternate?
  28. In how many ways \(4\) boys and \(3\) girls can be seated in a row so that they are alternate?
  29. In how many ways can the letters of the word “civilization” be rearranged?
  30. How many different words can be formed from the word “university” so that all vowels are together?
  31. In how many ways can the letters of the word “director” be arranged so that vowels are never together?
  32. How many words can be formed by rearranging the letter of the word “welcome”? How many of them end with ‘o’?
  33. How many words can be formed with the letters of the word “California” in such a way that vowels occupy vowels’ position and consonants occupy consonants’ position i.e. their relative order does not change?
  34. How many different words can be formed with the letters of the word “pencil” when vowels occupy even place?
  35. How many different words can be formed with five given letters of which three are vowel and two are consonants, no two vowels being together in any word?
  36. How many numbers greater than a million can be formed with the digits \(2, 3, 0, 3, 4, 2\) and \(3\) with no repetitions?
  37. In how many ways \(5\) Indians and \(4\) British can be seated at a round table if
    1. there is no restriction?
    2. all the four British sit together?
    3. all the four British do not sit together?
    4. no two British sit together?
  38. In how many ways \(5\) Indians and \(5\) British can be seated along a circle so that they are alternated?
  39. A round table conference is to be held between \(20\) delegates of \(20\) countries. In how many ways can they be seated if two particular delegates are always to sit together?
  40. Find the number of ways in which \(n\) different beads can be arranged to form a necklace.
  41. If \(^nP_4 = 12 \times ^nP_2,\) find \(n\).
  42. If \(^nP_5 = 20 \times ^nP_3,\) find \(n\).
  43. If \(^nP_4:^{n + 1}P_4 = 3:4,\) find \(n\).
  44. If \(^{20}P_r = 6840,\) find \(r\).
  45. If \(^{k + 5}P_{k + 1}= \frac{11(k - 1)}{2}.^{k + 3}P_k,\) find \(k\).
  46. If \(^{22}P_{r + 1}:^{20}P_{r + 2}=11:52,\) find \(r\).
  47. If \(^{m + n}P_2 = 90\) and \(^{m - n}P_2 = 30,\) find \(m\) and \(n\).
  48. If \(^{12}P_r = 11880,\) find \(r\).
  49. If \(^{56}P_{r + 6}:^{54}P_{r + 3} = 308000:1,\) find \(r\).
  50. How many numbers of four digits can be formed with the digits \(1, 2, 4, 5, 7\) with no repetitions?