Permutations and Combinations
If the new SAT is a haunted forest, permutation and combination
problems are the deepest, darkest, rarest trees. No, they are the
mysterious fluorescent fungus growing on those trees. Permutation
and combination problems are almost always hard, and most students
skip them because they take so long. But if you’re going for a Math
score above 700, you should know how to deal with them. And to deal
with permutations and combinations, you first have to know about
factorials. If you’re rushed for study time, though, and you’re
not trying to score a 700 on the Math section, this would be a good
section to skip.
The factorial of a number, represented by n!,
is the product of the natural numbers up to and including n:
The factorial of n is the
number of ways that the n elements
of a group can be ordered. So, if you become a wedding planner and
you’re asked how many different ways six people can
sit at a table with six chairs, the answer is 6! = 6 × 5
× 4 × 3 × 2 × 1 = 720.
Mutations are genetic defects that result in
three-headed fish. A permutation, however, is an ordering of elements.
For example, say you’re running for office in California, and there
are six different offices to be filled—governor, lieutenant governor,
secretary, treasurer, spirit coordinator, and head cheerleader.
If there are six candidates running, and the candidates are celebrities
who don’t care which office they’re elected to, how many different
ways can the California government be composed? Except that California
politics are funny, this question is no different from the question
about the ordering of six people in six chairs around the table.
The answer is 6! = 720 because there are six candidates
running for office and there are six job openings.
But, what if a terrible statewide budget crisis
caused three California government jobs to be cut? Now only the
three offices of governor, lieutenant governor, and spirit coordinator
can be filled. The same six candidates are still running. How many
different combinations of the six candidates could fill the three
positions? Time for permutations.
In general, the permutation, nPr,
is the number of subgroups of size r that can be taken
from a set with n elements:
For the California election example, you need to find 6P3:
Notice that on permutations questions, calculations become
much faster if you cancel out. Instead of multiplying everything
out, you could have canceled out the 3 × 2 × 1 in both numerator and denominator,
and just multiplied 6 × 5 × 4 = 120.
Permutations and Calculators
Graphing calculators and most scientific calculators have
a permutation function, labeled nPr.
Though calculators do differ, in most cases, you must enter n, then
press the button for permutation, and then enter r.
This will calculate a permutation for you, but if n is
a large number, the calculator often cannot calculate n!.
If this happens to you, don’t give up! Remember, the SAT never deals
with huge numbers: Look for ways to cancel out.
A combination is an unordered grouping of a set. An example
of a combination scenario in which order doesn’t matter is a hand
of cards: a king, an ace, and a five is the same as an ace, a five,
and a king.
Combinations are represented as nCr
where unordered subgroups of size r
from a set of size n
. Because the order of the
elements in a given subgroup doesn’t matter, this means that
will be less than nPr
Any one combination can be turned into more than one permutation.
is calculated as follows:
Here’s an example:
||Suppose six people
are running for three leadership positions, each of
which has the same duties and title. How many ways can this be done?
In this example, the order in which the leaders are assigned
to positions doesn’t matter—the leaders aren’t distinguished from
one another in any way, unlike in the California government example.
This distinction means that the question can be answered with a
combination rather than a permutation. So, to figure out how many different
groups of three can be taken from a group of six, do
There are only 20 different ways to elect
three leaders, as opposed to 120 ways when the leadership
jobs were differentiated.
Combinations and Calculators
As with permutations, there should be a combination function
on your graphing or scientific calculator labeled nCr.
Use it the same way you use the permutation key.
That’s it, everything, the whole SAT mathematical banana—from
Numbers and Operations to Data, Statistics, and Probability. You’ve
now covered every little bit of math that might appear on the SAT.
To make your job even easier, at the start of this section is a
chart that summarizes the most important SAT math facts, rules,
and formulas for quick reference and easy studying.