Combinations (Making sets, combining only a few options)

Combinations get trickier when you don’t have to choose one item from each group.  If you are ordering lunch and can choose among three sandwiches, three types of chips, and three drinks, and you must choose one of each, you multiply your options together and get 27 possible combinations (for more, see the lesson on Combinations (Making sets, choosing one of each option)).

If, when you are making your combinations, you are allowed to choose from only some (not necessarily all) categories, the possible combinations grow considerably (and the process of figuring out how many combinations there are) gets more complicated.

Combinations are possible combinations of items.  There are math formulas for figuring out the number of possible combinations, but there are two types of problems for which it’s easier to draw out all of the possibilities:

• When your combination can have varying numbers of events (e.g., either 1 topping, two toppings, or three toppings on a sundae). In this case, your best option is to map out all of the possibilities (and look for patterns if there are a lot of combinations).
• If you have a limiting factor on your combinations (e.g., the numbers must add up to a certain amount).  In this case, when you draw out the options, you can see them all and eliminate the ones that don’t fit the criteria.

What's the trick to drawing these problems out?

Be methodical.  Do not write down combinations randomly.  Create order in your notes.  Start with the options that include the fewest number of options, then work your way up (the single options are so obvious, you're certain to get those right, so start there!).  Try to combine items in the same order each time.  If you follow these "order" rules, you will be less likely to miss a possible combination.  Also, look out for duplicates!  These will mess you up too.

Example: Luis can select one or more of the following 3 toppings for his ice cream: nuts, whipped cream, and cherries.  If he selects one or more, how many different combinations of toppings are possible? (Assume that the order of the toppings does not matter.)

First, map out the single outcomes:

Nuts only

Cream only

Cherries only

That's three possible sundaes.

Next, map out the double outcomes (Make sure to pair in order (e.g., nuts with everything, then cream with everything) to make sure you don't skip anything:

Nuts & cream

Nuts & cherries

Cream & nuts

Cream & cherries

Cherries & nuts

Cherries & cream

Before you count, cut out the duplicates!

Nuts & cream

Nuts & cherries

$\require{cancel}\cancel{\text{Cream & nuts}}$

Cream & cherries

$\cancel{\text{Cherries & nuts}}$

$\cancel{\text{Cherries & cream}}$

If you look at the pattern here, you'll see that, if you follow the same order, each time you list a new item, there will be fewer options (because some will have already been made).

There are 3 possible sundaes with 2 items.

Finally, map out all of the triple items:

Nuts, cream, & cherries

There is one possible sundae with all three options.

Count up all of your sundaes: $3+3+1=7$

There are 7 possible sundaes.

Overall, as annoying as it can be, sometimes the easiest way to find combinations is to draw them out.  The smartest thing to do is to write them out methodically and in order.  You'll see the patterns, be able to catch your mistakes, and count them up.