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Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

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BuyFindarrow_forward

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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At a certain company, passwords must be from 3 5 symbols long and composed from the 26 uppercase letters of the Roman alphabet, the ten digits 0 9 . and the 14 symbols ! , @ , # , $ , % , ^ , & , * , ( , ) , , + , { and }.

a. How many passwords are possible if repetition of symbols is allowed?
b. How many passwords contain no repeated symbols?
c. How many passwords have at least one repeated symbol?
d. What is the probability that a password chosen at random has at least one repeated symbol?

To determine

(a)

To find how many different license plates can the state produce.

Explanation

Given information:

License plates consist of from zero to three letters followed by from zero to four digits, with the provision, however, that a blank plate is not allowed.

Concept used:

A license plate may contain either zero or one or two or three letters.

Calculation:

The objective is to determine the number of license plates that a state can produce.

Determine the total number of license plates that consists of zero to three letters.

That is, a licensee plate may contain either zero or one or two or three letters.

If the license plate contains zero letters, then there is only one possibility.

If the license plate contains one letter, then the letter can be chosen in 26 ways.

So, total number of license plates which contains only one letter is 26.

If the license plate contains two letters, then first letter can be filled in 26 ways and second letter can also be filled in 26 ways as repetition is allowed.

So, total number of license plates which contains two letters is 26×26=676.

If the license plate contains three letters, then first letter can be filled in 26 ways, second letter in 26 ways, and third letter in 26 ways.

So, total number of license plates, which contains three letters is 26×26×26=17576

Hence, total number of license plates that consists of zero to three letters is 1+26+676+17,576=18,279.

Determine the total number of license plates that consists of zero to four digits.

That is, a license plate may contain either zero or one or two or three or four digits.

There are 10 digits, namely 0,1,.....,9.

If the license plate contains zero digits, then there is only one possibility.

If the license plate contains one digit, then the digit can be chosen in 10 ways.

So, total number of license plates, which contains only one digit, is 10

To determine

(b)

To find how many different license plates can the state produce.

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