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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.

Solution not provided.

Ans. (a) 36000 (b) 14400

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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?

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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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