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Permutations and Combinations

Short Answer Type

101. Prove that : (i) P(n, n) = P(n, n – l) (ii) P (n, r) = (n – r + 1) P (n, r – 1)

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102. If: ^{n– 1} P₃ : ^{n + 1}P₃ = 5 : 12, then find n.

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103.

In how many ways can the letters of the word ‘PENTAGON’ be permuted, using all letters ?

(a) How many of them begin with P and end with N ?

(b) How many of them have the three vowels together ?

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104. The letters of the word ‘TUESDAY’ are arranged in a line, each arrangement ending with letter S. How many different such arrangements are possible ? How many of these begin with D?

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105. How many words with or without meaning can be formed using letters of word ‘DAUGHTER’ so that vowels and consonants occur together ?

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106.

Find the number of words formed by arranging the letters of the word ‘DAUGHTER’ so that

(a) not all vowels are together (b) no two vowels are together.

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107. How many 3 letter words can be formed from the letter of the word ‘ORIENT’ ? How many of these words contain T?

Solution not provided.

Ans. 120; 60

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Long Answer Type

108. In how many ways can 5 boys and 3 girls be seated in a row so that

(a) the girls are together? (b) all the three girls are not together?

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Short Answer Type

109. How many 3-digit numbers can be formed using digits 1, 2, 3, 4, 5 if no digit repeated? How many of these will be even?

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110. From a committee of 8 persons, in how many ways can we choose a chairman and a vice-chairman assuming one person cannot hold more than one position?

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