How Mathematicians Count

When there are m ways to do one thing, and n ways to do another,
then there are m×n ways of doing both.

There are 6 flavors of ice-cream, and 3 different cones.
That means 6×3=18 different single-scoop ice-creams you could order.

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1) How many 5 letter PASSWORDS using
all of these letters (only once): A, Q, C, D, F

2) How many 3 letter PASSWORDS using
all of these letters (only once): A, Q, C, D, F

3) How many 5 letter PASSWORDS using
all of these letters (Letters CAN REPEAT): A, Q, C, D, F

4) Choose 3 ingredients from a list of 5 toppings.
How many styles of pizza can I make?

5) How many pizzas with 4 crusts to choose from
and 1 ingredient form a list of 5 toppings?

6) I know my computer password has the 6 letters
in "BANANA". How many possible rearrangements
of these letters is possible?


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"How many ... can be formed from a list of say 5 items?"
This type problem requires one to decide on a method:
Tree diagram
Multiply 5 times 5 times 5 ...
 - use 5!
 - use Combinations nCr
 - use Permutations nPr
 - use 5! but divide by 2! times 2!
(as in 5 letter words from AABBC)
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How many three digit numbers can be formed with the digits: 1, 2, 3, 4, 5?
A) If each digit can only be used once then,

5 P 3 = (5!) divided by (5-3)! = 5!/2! = 60
B) If a digit can be repeated then,

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Permutations with Repetition of Indistinguishable Objects:
Indistinguishable objects are simply items (letters) that are repeated in the original set.
For example, if the word MOM was used the two letter M's
are indistinguishable from one another, since they repeat.
So our normal method that gives us 3!  =  6 ways needs to be adjusted.
If we are looking for answers that are not duplicates (unique answers),
we must deal with any letters (objects) that repeat in the original set.                        
In general, repetitions are taken care of by dividing the permutation by (2!)
the factorial of  the number of objects that are identical.  

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How many different 5-letter words
can be formed from the word   "APPLE"  ?

(You divide by  2!  because the letter  P  repeats twice.)
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How many different six-digit numerals (6 digit STRINGS)
can be written using all of the following six digits:
       4,4,5,5,5,7    ?

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The signal mast of a ship can raise nine flags at one time
 (three red, two blue and four green).
How many different signals can be communicated
by the placement of these nine flags?
The order of the elements does matter.
9! divided by (3!)(2!)(4!) = 288 signals
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