Many probability problems revolve counting - there are some techniques we can use to help "count correctly". They are:

1) The Multiplication Principle

2) Permutations

3) Combinations

Solution: Consider a tree diagram of this situation - here are all possible outcomes listed:

To do this problem, consider that for each way you can select a shirt, you have two pairs of pants. So you have 3 * 2 = 6 different outfits available to you. This is an application of the Multiplication Principle. Here is the formal statement of this rule

Another multiplication principle problem

How many different license plates in California? (discounting personalized plates)

Each plate starts with a digit from 1-9 followed by three letters (repeats okay) then followed by three digits from 0-9 (repeats okay). How many plates? How many plate without any repetition of letters or digits?

Solution:

To determine how plates are available, consider that the first item on the license plate is a digit from 1-9, you have nine choices for the first item. Then for each of these 9 choices you have 26 choices for EACH of the next three spaces (repeats of letters are okay) and finally you have 10 choices for the last 3 spaces (again repeats are okay). So calculating this out - you get:

9*26*26*26*10*10*10 = 158184000 ways

If no repeats of digits or letters are allowed then you still have 9 choices for the first space and 26 for the second, then you have 25 for the next letter (can't repeat the one already chosen ) and 24 for the next letter (again you can't repeat the first two chosen). The same holds for the digits in the last three spaces (9, 8 and 7 possibilities here - note that one digit has already been chosen for the first space so you can use that one)

9*26*25*24*9*8*7=70761600 ways

Another multiplication principle problem:

You have 7 different books on a shelf. In how many ways can they be arranged?

Solution: There are 7 choices for the first book, 6 for the second, 5 for the third, etc.. So you get:

7*6*5*4*3*2*1=7!

This is called 7 factorial. The ! in mathematics has special meaning

Notation: n! = n*(n-1)*(n-2)*(n-3)*... 1 read as n factorial

Factorial's get large quickly try on your calculator - 10!, 100! 1000!, 10000!. Bet some of these give you and error - they get too large for the display or memory of the machine

The multiplication principle works well for many problems - but not all!

Here we can think about this problem as selecting three from the group and order of selection is important. If you are selecting from a group and order matters you use the permutation formula

Where n is the total number you are selecting from and r is the number of items you are selecting. In our example above this n=15, r =3

So . You calculator has a button (probably) for this operation

another to try:

Most permutation problems can be done with multiplication principal - here is one

You have seven different colored flags that you can use to hoist up a flagpole. In how many different ways can you arrange 3 of these flags on the flagpole?

Solution: We can consider this problem in two ways. First we can think of that we are selecting from 7 flags to hoist up first, so we have 7 choices for the first position. Then we have 6 choices for the next position and 5 for the final position. This gives 7*6*5=210 ways. Here we used the multiplication principle.

If instead we think of taking 3 flags from the group of seven, we can use a permutation.

In permutations - order was important, what if order is not important

Instead of electing a president, vice-president and secretary you elect a ruling committee of 3. In how many ways can this be done?

Solution: Since if you are selected to be on a committee, it does not matter if order is important, we can't use a permutation. We use a combination. Here is the formula for this:

Notice that we divide by r!. To see why -consider the example of selecting a committee of 3 from 15 people. Suppose Jack (represented by J), Debra (represented by D) and Nick (represented by N) are selected to be on the committee. If order did matter then the following groups would represent different outcomes:

JDN

JND

NJD

NDJ

DNJ

DJN

But if this is a committee then these are all the same committee. Since there are 3 people being selected this means any three items can be arranged in 3! ways. So in the case of selecting r items, we divide by r!

Here is another combination problem:

At Pete's pizza palace - they have the following toppings to choose from

In how many ways can yon make a three topping pizza?

Solution: does order matter? I don't think so. If you order a sausage, mushroom and onion pizza, does it matter in which order they put these items on the pizza? Most people do not think so. So Let's use a combination:

ways!