4.15 SAT Math - Combinations

Combinations

  Combinations, like probability, can have independent or dependent choices.  Take a look at these:

  Five employees are vying for spots on a 3-person committee.  How many different 3-person committees can be formed?

  Five employees are vying for spots on a 3-person committee.  There will be a Manager, an Editor, and a Secretary.  How many different 3-person committees can be formed?
  Tom is making a themed meal for his dinner party.  If his cookbook offers him the choices of 4 appetizers, 4 main dishes, and 3 desserts, how many different 3-course meals can he make?

In the first question, the outcome of each choice is dependent on the earlier.  Let’s say you assign letters to each employee, calling them A, B, C, D and E.  If A makes the committee, there are now only 4 people left that can serve under the other two positions, and A can’t serve in two slots.  You can start to make a list of all the people that could serve with A like so:

ABC
ABD
ABE
ACD
ACE
ADE

From there, what if A doesn’t make it?  Make a list of options starting with B:

BCD
BCE
BDE

What about if neither A nor B make it?

CDE

In these cases, the order the people are chosen makes no difference because each position is equal.  For questions in which order does not matter, the simplest way to solve is to simply write out all your options, keeping the list neat and organized to be sure you catch all the possibilities.  Count the number of combinations, and you have your answer.

Starting SAT Score 700 or Above:

If order does not matter, you can use factorials as a shortcut.  Set up a fraction in which the numerator consists of the number in each option multiplied together; the denominator will be the choices field factorial.  Using the above example, there are 5 people to choose from for the first spot, 4 for the second, and 3 for the 3rd.  There are 3 spots available.  So your equation will be , giving you an answer of 10.

In the second question, “Five employees are vying for spots on a 3-person committee.  There will be a Manager, an Editor, and a Secretary.  How many different 3-person committees can be formed?” order does matter, because employee A could be Manager, or Editor, or Secretary.  Each spot is a different position.  You could begin by writing out your possibilities as before:

ABC
ABD
ABE

But when it comes to the AC group, B is now an option again.

ACB
ACD
ACE

Same with AD:

ADB
ADC
ADE:

And AE:

AEB
AEC
AED

We know have 12 possibilities just with A as the manager.  If B is chosen as the manager, will we have the same number of possibilities?  Yes, so we can simply multiply the number of possibilities for A times the number of people: 12 • 5 = 60.

However, there’s an even shorter way to do it when order matters.
How many possibilities are there for Manager at the beginning?  5.  One of those five will be chosen, leaving how many possibilities for Editor?  4.  Which leaves only 3 possibilities for Secretary.  We can simply multiply 5 • 4 • 3 to get the answer of 60 combinations.

The last question, Tom is making a themed meal for his dinner party.  If his cookbook offers him the choices of 4 appetizers, 4 main dishes, and 3 desserts, how many different 3-course meals can he make? is an example of a separate pool of options.  What does this mean?  This means that it may appear that order matters, but in fact, the choice from one category does not affect a choice from another.  So how many options does he have for appetizers? 4.  How many for main dishes? 4.  How many for desserts?  3.  Multiply them all together to get 4 • 4 • 3 and an answer of 48 combinations.

If you’re not sure which shortcut to use on the SAT, it’s always safer to write out combinations, but be careful!  Combinations are mostly on this test for the purpose of wasting your time.  They are meant to take time away from other problems that may be easier to solve.  Every question is only worth 1 point, easy or hard.  Don’t get stubborn about getting an answer if it’s taking you too long.  Skip it and move on.