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Permutations and Combinations

Short Answer Type

81. How many words can be formed by using the letters of the word ‘ORIENTAL’ so that A and E always occupy the odd places?

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82.

In how many ways can 5 girls be seated in a row so that two girls Ridhi and Sanya are:

(a) always together (b) never together

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83. How many different numbers of 6-digits can be formed by using digits 4,5,6,7,8,9 (a) no digit being repeated (b) digits may be repeated?

Also, find in case (a) as to how many of them arc odd, when repetition is not allowed?

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84. How many different numbers of 5-digit can be formed by using the digits 2,3, 4, 5 and 7, without repetition, so that the number is not divisible by 5?

Number of digit available = 5 (all distinct)

Number of digits to be used = 5

Since the number is not divisible by 5.

$rightwards double arrow$ 5 cannot occupy box 5.

Number of permutations for box $5 equals straight P presuperscript 4 subscript 1 space equals space fraction numerator 4 factorial over denominator 3 factorial end fraction equals 4$ ...(i)

Now, we have 4-digits and 4 boxes.

Number of permutations = $straight P presuperscript 4 subscript 4 space equals space 4 factorial space equals space 24$ ... (ii)

From (i) and (ii), the total number of permutations = 4 x 24 = 96

Hence, the number of numbers formed, that are not divisible by 5 = 96.

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85. In how many ways can three prizes be given to 10 boys when a boy may receive any number of prizes?

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86. Find the number of 4-digit numbers in which any digit may be repeated any number of times.

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87. Find the number of 4-digit numbers in which no digit is repeated.

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88. Find the number of 4-digit numbers in which at least one digit is repeated.

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Long Answer Type

89. There are three prizes to be distributed among 6 students. In how many ways can this be done when

(a) no boy gets more than one prize.

(b) there is no restriction as to the number of prizes that a boy may get.

(c) no boy sets all prizes.

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90.

There are 8 students appearing for an examination, of which 3 appear in mathematics paper, and 5 in other different subjects. In how many ways can they be seated if

(a) all the students appearing for mathematics paper are together.

(b) the students appearing for mathematics paper are not all together.

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