Math Homework help? ( product tables and arrangements)?

I have two problems for my mathematical perspectives class that I don't get.

The first one I have to make a product table and answer a,b,and c.

1. If we roll 2 six-sided dice (numbered 1 to 6), we have 36 possibilities. Build a Product Table for the 2 dice and tell:

a) The number of dice rolls where neither number is Even

b) The number of rolls where both numbers are Perfect Squares

c) The number of rolls where one number is exactly twice the other number

The second problem has to do with arrangements.

2. We have to create passwords for a new computer system.  We can use any of 5 letters (a, b, c, d, e) in both upper and lower case and any of 4 digits (0, 1, 2, 3).

a) How many 3-character passwords can we create?

b) How many 3-character passwords can we create if we cannot repeat any character?

c) Oops!  The new system is case-insensitive (it cannot tell upper case and lower case apart)!  How many 3-character passwords do we now have?

Help with either would be much appreciated!! Thank you!!

Relevance
• 1 month ago

I'm not sure what is meant by a "Product Table" in this case:

Is it a table of the product of the two numbers rolled,

or a table of the number pairs produced?

The first would be the normal meaning of "product,"

but the second makes far more sense, given the following questions,

so that's what I'll assume is meant here.

\ .... 1 .... 2 .... 3 .... 4 .... 5 .... 6 ... <-- first die

1 . 1-1 . 2-1 . 3-1 . 4-1 . 5-1 . 6-1

2 . 1-2 . 2-2 . 3-2 . 4-2 . 5-2 . 6-2

3 . 1-3 . 2-3 . 3-3 . 4-3 . 5-3 . 6-3

4 . 1-4 . 2-4 . 3-4 . 4-4 . 5-4 . 6-4

5 . 1-5 . 2-5 . 3-5 . 4-5 . 5-5 . 6-5

6 . 1-6 . 2-6 . 3-6 . 4-6 . 5-6 . 6-6

|

second die

a) The number of dice rolls where neither number is Even

Half the numbers rolled are even, and half (three) are not.

Just as the table for six possible numbers has 6*6=36 entries,

the table for three possible numbers (where neither is even)

has 3*3=9 entries, and is a subset of the table shown.

b) The number of rolls where both numbers are Perfect Squares

1 and 4 are the perfect squares, so that would be a 2 by 2 subset

of the table, and 2*2 = 4 entries.

They are, of course, the entries 1-1, 1-4, 4-1, and 4-4.

c) The number of rolls where one number is exactly twice the other number

They'd have to be 1 and 2 (which occur as 1-2 and 2-1),

2 and 4 (which occur as 2-4 and 4-2),

and 3 and 6 (which occur as 3-6 and 6-3),

for a total of 6 such rolls.

2. We have to create passwords for a new computer system. We can use any of 5 letters (a, b, c, d, e) in both upper and lower case and any of 4 digits (0, 1, 2, 3).

a) How many 3-character passwords can we create?

We have five upper-case letters, five lower-case letters, and four digits to use.

So each character in the password can be any of 14 characters,

and there are

14^3 = 2,744 possible passwords.

b) How many 3-character passwords can we create if we cannot repeat any character?

We can have any of 14 choices for the first character,

which will leave 13 choices for the second,

which in turn will leave 12 choices for the third.

So that's

14*13*12 = 2,184 possible passwords.

c) Oops! The new system is case-insensitive (it cannot tell upper case and lower case apart)! How many 3-character passwords do we now have?

That takes away 5 possibilities for each character,

so there are

9 * 8 * 7 = 504 if we cannot repeat a character,

or

9^3 = 729 if we are allowed repeated characters.