63. Permutations and Combinations Problems Part 2

51. How many numbers of \(5\) digits can be formed with the digits \(0, 1, 2, 3, 4?\)

52. How many numbers between \(100\) and \(1000\) can be formed with the digits \(1, 2, 3, 4, 5, 6, 7;\) with no digits being repeated?

53. How many numbers between \(100\) and \(1000\) can be formed with the digits \(0, 2, 3, 4, 8, 9\); with no digits being repeated?

54. Find the total no. of nine digit numbers which have all different digits.

55. How many numbers between \(1000\) and \(10000\) can be formed with the digits \(0, 1, 2, 3, 4, 5\); with no digits being repeated?

56. How many different numbers greater than \(5000\) can be formed with the digits \(0, 1, 5, 9\); with no digits being repeated?

57. Find the number of numbers between \(300\) and \(4000\) that can be formed with the digits \(0, 1, 2, 3, 4, 5\); no digits being repeated.

58. If repetition of digits is not allowed, how many numbers of four digits divisible by \(5\) can be formed with the digits \(0, 4, 5, 6, 7\)?

59. How many different number of \(6\) digits each can be formed with the digits \(4, 5, 6, 7, 8, 9\)? How many of them are not divisible by \(5\); with no digits being repeated?

60. How many even numbers of \(5\) digits can be formed with the digits \(1, 2, 3, 4, 5\)?

61. How many numbers less than \(1000\) and divisible by \(5\) can be formed in which no digit repeats.

62. How many numbers between \(100\) and \(999\) can be formed with the digits \(0, 4, 5, 6, 7, 8\)? How many of them are odd?

63. Find the number of even numbers that can be formed with the digits \(0, 1, 2, 3, 4\); with no digits being repeated.

64. Find the number of numbers of six digits with the digits \(1, 2, 3, 4, 5, 6\) in which \(5\) always occupy in the tens place with no digits being repeated.

65. A number of four different digits is formed by using the digits \(1, 2, 3, 4, 5, 6, 7\). (i) Find how many such numbers can be formed? (ii) How many of them are greater than 3400?

66. Find the number of numbers of \(4\) digits formed with the digits \(1, 2, 3, 4, 5\) in which \(3\) occurs in the thousand’s place and \(5\) occurs in the unit’s place.

67. Find the number of numbers of \(4\) digits formed with the digits \(0, 1, 2, 3, 4, 5\); with no digits being repeated. How many of these are greater than \(3000\)?

68. How many number of numbers can be formed by using any number of digits \(0, 1, 2, 3, 5, 7, 9\)?

69. How many different numbers can be formed with the digits \(1, 3, 5, 7, 9\); when taken all at a time and what is their sum?

70. Find the sum of all four digit numbers that can be formed with the digits \(3, 2, 3, 4\).

71. Find the sum of all numbers greater than \(10,000\) formed with the digits \(0, 2, 4, 6, 8\); with no digits being repeated.

72. Find the sum of \(5\) digit numbers with the digits \(3, 4, 5, 6, 7\); no digits being repeated.

73. Find the sum of all \(4\) digit numbers that can be formed with \(0, 2, 3, 5\).

74. A servant has to post \(5\) letters and there are \(4\) letter boxes. In how many ways he can post the letters?

75. In how many ways can \(3\) prizes be given to \(5\) students when each student is eligible for any number of prizes?

76. In how many ways can \(n\) things be given away to \(p\) persons? Each person can get any number of things (\(n>p\)).

77. There are \(m\) men and \(n\) monkeys (\(m<n\)). If a man may have any number of monkeys, in how many ways every monkey have a master?

78. In how many different ways the following \(5\) prizes can be given to \(10\) students? First and second in Mathematics; first and second in Chemistry and first in Physics.

79. There are stalls for \(12\) animals in a ship. In how many ways the shipload can be made if there are cows, calves and horses to be transported with each being \(12\) in number.

80. In how many ways can \(5\) delegates be put in \(6\) hotels of a city if there is no restriction?

81. Find the number of numbers of \(5\) digits that can be formed with the digits \(0, 1, 2, 3, 4\) if repetition of digits is allowed.

82. In how many ways rings of \(6\) different types can be had in \(4\) fingers?

83. Find the number of \(4\) digit numbers greater than \(3000\) that can be formed with the digits \(0, 1, 2, 3, 4, 5\) if repetition of the digits is allowed.

84. In a town the care plate numbers can be of three or four digits without digit \(0\). What is the maximum number of cars that can be numbered?

85. In how many ways can a ten question multiple choice examination with one correct answer be answered if there are four choices to each question? If no two consecutive questions are answered the same way, how many ways are there?

86. There are two books each of three volumes and two books each of two volumes. In how many ways can the ten books be arranged on a table so that the volumes of the same book are not separated?

87. A library has \(5\) copies of one book, \(4\) copies of \(2\) books, \(6\) copies of \(3\) books and single copy of \(8\) books. In how many ways all books can be arranged in so that copies of the same book are always together?

88. In a dinner party there are \(10\) Indians, \(5\) American and \(5\) British. In how many ways they can be seated so that all persons of same nationality always sit together?

89. In a class there are \(4\) girls and \(6\) boys. In how many ways they can be seated in a row so that no two girls are together?

90. Show that the number of ways in which \(n\) books can be arranged on a shelf so that two particular books shall not be together is \((n - 2) (n - 1)!\)

91. You are given six balls of different colors(black, white, red, green, violet, yellow). In how many ways can you arrange them in a row so that black and white balls may never come together?

92. In how many different ways can \(15\) I.Sc. and \(12\) B.Sc. students be arranged in a line so that no two B.Sc. students occupy consecutive positions?

93. In how many ways can \(18\) white and \(19\) black balls be arranged in a line so that no two white balls may be together? It is given that balls of same color are identical.

94. Show that the number of ways in which \(p\) positive and \(n\) negative signs may be placed in a row so that no two negative signs may be together is \(^{p + 1}C_n\)

95. \(m\) men and \(n\) women are to be seated in a row so that no two women sit together. If \(m>n\), then show that the number of ways in which then can be seated is \(\frac{m!(m + 1)!}{(m - n + 1)!}\)

96. \(3\) women and \(5\) men are to sit in a row. Find in how many ways they can be arranged so that no two women sit next to each other.

97. Find the number of ways of arranging the letters \(a, a, a, a, a, b, b, b, c, c, c, d, e, e, f\) in a row, if letters \(c\) are separated from one another.

98. Find the number of different permutations of the letters of the word “Banana”.

99. How many words can be formed from the letters of the word “circumference” taken all together.

100. There are three copies of each of four different books. In how many ways they can be arranged in a shelf?