Many reallife problems involve counting the number of ways that things can be chosen.
In making the choices, sometimes order matters, and sometimes it doesn't.
For example, suppose you're ordering a $3$topping pizza.
Whether you say ‘Pepperoni, mushrooms, and onions, please’
or ‘Yoo hoo! Make that mushrooms, pepperoni, and onions’
you'll get the same pizza.
Here, order doesn't matter.
Or, suppose you're choosing numbers and letters for a license plate.
The license plate ‘KC 157’ is different from ‘CK 751’,
even though they use the same letters and numbers.
Here, order matters.
Here's another way to think about ‘order matters’ versus ‘order doesn't matter’.
Suppose you have three letters ($\,\text{A}\,$,
$\,\text{B}\,$, and $\,\text{C}\,$) in an urn.
You are told to choose two letters, without replacement.
That is, you'll choose a letter, not return it to the urn, and then choose another.
How many ways are there to do this?
Well, it depends.
Does order matter, or not?
If order doesn't matter, then there are only three possible choices:
you could walk away with the set
$\,\{\text{A},\text{B}\}\,$
or the set
$\,\{\text{A},\text{C}\}\,$
or the set
$\,\{\text{B},\text{C}\}\,$.
If order does matter, then each of the sets above could be arranged in two different ways,
giving six possible choices:
$\,\text{AB}\ \ \text{BA}\ \ \text{AC}\ \ \text{CA}\ \ \text{BC}\ \ \text{CB}$
Mathematicians use two different words, depending on if order matters, or if it doesn't:
For example, there are six permutations when choosing two letters from
$\,\text{A}\,$, $\,\text{B}\,$, and $\,\text{C}\,$:
$$
\cssId{s31}{\text{AB}\ \ \ \ \text{BA}\ \ \ \ \text{AC}\ \ \ \ \text{CA}\ \ \ \ \text{BC}\ \ \ \ \text{CB}}
$$
For example, there are three combinations when choosing two letters from
$\,\text{A}\,$, $\,\text{B}\,$, and $\,\text{C}\,$:
$$
\cssId{s37}{\{\text{A},\text{B}\}}\ \ \ \ \
\cssId{s38}{\{\text{A},\text{C}\}}\ \ \ \ \
\cssId{s39}{\{\text{B},\text{C}\}}
$$
This author likes to report combinations as sets,
to emphasize the fact that order doesn't matter.
(Recall that when using list notation for a set, the order that the members are listed doesn't matter.)
Think of a combination as a bunch of items that are thrown into a basket.
All that matters is what ends up in the basket; it doesn't matter how they got in there.
Whenever you ask a question about combinations or permutations,
there are usually two numbers that are important:
Returning to the experiment of choosing two letters from the urn containing $\,\text{A}\,$, $\,\text{B}\,$, and $\,\text{C}\,$:
$$
\cssId{s78}{ {}_3C_2}
\cssId{s79}{= \binom{3}{2}}
\cssId{s80}{= 3}
\cssId{s81}{\ \ \text{ and }}\ \
\cssId{s82}{ {}_3P_2 = 6}
$$
Formulas for both $\,{}_nC_k\,$ and $\,{}_nP_k\,$ will be derived in a
future section.
In this web exercise,
the emphasis is on understanding the concept of ‘order matters’
versus ‘order doesn't matter’,
and on getting comfortable with the notation.
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
