Combinatorics | Number Theory Answers

In a certain country telephone numbers have 9 digits. The first two digits are the area code (03) and are the same within a given area. The last 7 digits are the local number and cannot begin with 0. How many different telephone numbers are possible within a given area code in this country?

To buy a computer system, a customer can choose one of 4 monitors, one of 2 keyboards, one of 4 computers and one of 3 printers. Determine the number of possible systems that a customer can choose from.

Use the method of contradiction to prove that √5 is not a rational number.

Let p(x): x is mammal and q(x): x is animal. Translate the following in English: (∀x)(q(x)∧(∼p(x))).

a. In how many different ways you put 20 equal balls, into two numbered boxes?

c. What if you have 20 equal balls, three numbered boxes, and you want to place at least 2 balls in the first box, and no more than 10 balls in the last box?

A password for the computer is a string of digits and/or letters. The password must obey the following rules:

a. The length of the password must be exactly 8 characters

b. The password must contain at least one digit, and no more than three digits.

What is the number of different passwords you can form?

There are 5 chairs in a row, and three people. In how many ways can you place the people on the chairs? (each person must occupy exactly one chair, so that two chairs will remain free.)

How many words of length 8 can you form where the first letter is same as the last letter?

How many words of length 10 can you form where no letter is repeated?

How many words of 10 letters can you form using the Italian alphabet? (there are 21 different letters in the Italian alphabet or which 5 are vowels.)