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Discrete Mathematics and its Applications
Found in: Page 439
Discrete Mathematics and its Applications

Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

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Short Answer

Explain how to find the number of bit strings of length not exceeding 10 that have at least one 0 bit.

The number of string of length ≤10 having at least 0 bit is 2036 .

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Definition of Concept

Permutations: A permutation of a set is a loosely defined arrangement of its members into a sequence or linear order, or, if the set is already ordered, a rearrangement of its elements, in mathematics. The act of changing the linear order of an ordered set is also referred to as "permutation."

Lexicographic order: The lexicographic or lexicographical order (also known as lexical order or dictionary order) in mathematics is a generalization of the alphabetical order of dictionaries to sequences of ordered symbols or, more broadly, elements of a totally ordered set.

Step 2: Explain how to find the number of bit strings of length not exceeding 10 that have at least one 0 bit

Considering the given information:

Length of string ≤10 having at least one 0 bit.

Using the following concept:

Number of bit string of length n=2n

The number of bit strings of length ≤10 can be calculated as:

=21+22+23+24+25+26+27+28+29+210=2+4+8+16+32+64+128+256+512+1024=2046

Number of strings without zeroes of length 1=1

Number of strings without zeroes of length 2=1

Number of strings without zeroes of length 3=1

Number of strings without zeroes of length 10=1

As a result, the number of bit strings of length ≤10 with at least one 0.

=2046-10

20336

Therefore, the required number of string of length ≤10 having at least 0 bit is 2036 .

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One hundred tickets, numbered \(1,2,3, \ldots ,100\), are sold to \(100\) different people for a drawing. Four different prizes are awarded, including a grand prize (a trip to Tahiti). How many ways are there to award the prizes if

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